Is sound wave a transverse wave?

AI Thread Summary
Sound waves are primarily longitudinal in fluids like air, as these mediums do not support shear forces. In solids, sound can be both longitudinal and transverse, with longitudinal waves traveling through the material and transverse waves occurring at the surface. While the ear primarily detects longitudinal vibrations, surface waves can also generate sound when they interact with air. The discussion highlights the complexity of defining sound, as it can vary depending on the medium and context. Ultimately, sound in solids can be both longitudinal and transverse, but only longitudinal waves are typically recognized as sound in air.
jerry0696
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i had always belived that sound wave was a longitudinal wave till i come across something saying that its a transverse wave in solid.can anyone explain is it so??
 
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jerry0696 said:
i had always belived that sound wave was a longitudinal wave till i come across something saying that its a transverse wave in solid.can anyone explain is it so??

In a fluid like air, sound waves are only longitudinal, since fluids don't support shear forces. Solids support both longitudinal and transverse sound waves, and they typically travel at different speeds.
 
phyzguy said:
In a fluid like air, sound waves are only longitudinal, since fluids don't support shear forces. Solids support both longitudinal and transverse sound waves, and they typically travel at different speeds.

if solid support both does that mean that sound is a transverse wave in solids??
 
jerry0696 said:
if solid support both does that mean that sound is a transverse wave in solids??

There are longitudinal sound waves in solids.

There are also transverse sound waves in solids.
 
jerry0696 said:
does that mean that sound is a transverse wave in solids??

Sound can be a transverse wave in a solid, but it doesn't have to be transverse. It can be longitudinal instead, like in a gas or liquid.
 
jtbell said:
Sound can be a transverse wave in a solid, but it doesn't have to be transverse. It can be longitudinal instead, like in a gas or liquid.

but is sound transverse ,longitudinal or both in solid??
everyone is telling me that sound can only be longitudinal (my teachers)
 
Inside a solid, sounds is longitudinal. However, at the surface of a solid, sound is transverse. The surface is a like a membrane, its oscillations can only be transverse.
 
However, at the surface of a solid, sound is transverse. The surface is a like a membrane, its oscillations can only be transverse

Am desperately trying to picture that

any further info please ?

thanks
Dave
 
  • #10
davenn said:
Am desperately trying to picture that

any further info please ?

It is almost true by definition, for a common sense definition of "sound". If the surface of the solid doesn't move transversely to the surface, it doesn't transmit any pressure vibrations to the air. (OK, let's ignore the tiny amount of energy that would get into the boundary layer and be dissipated by the air viscosity before it got anywhere else.)

If course a solid can vibrate with no transverse motion at the surface, for example torsional vibrations of a disc on a shaft, but that type of vibration doesn't generate "sound".
 
  • #11
voko said:
Inside a solid, sounds is longitudinal. However, at the surface of a solid, sound is transverse. The surface is a like a membrane, its oscillations can only be transverse.

you are telling me that it is a longitudinal wave in the solid but transverse wave at the surface.
do you know any book that can give me further explanation on this topic??
 
  • #12
I think the answer to the OPs question is no. Sound waves are longitudinal and not transverse. By definition sound waves are related to those types of waves that can be sensed by the ear and the ear drum reacts to longitudinal vibrations.
A solid may be able to vibrate in different ways but only the longitudinal components of those vibrations can be transferred to any surrounding air as sound waves. It reminds me of the longitudinal and transverse vibrations that can be set up on a stretched string. Both set up longitudinal vibrations (sound waves) in the surrounding air.
 
  • #13
  • #14
There are different modes of vibration in solids but only those that can set up longitudinal waves in the surroundings are related to sound.
 
  • #15
Dadface said:
There are different modes of vibration in solids but only those that can set up longitudinal waves in the surroundings are related to sound.

If you attach a solid body directly to an acoustic sensor, a transverse wave will excite it just the same as a longitudinal wave would.

If there is an air gap, transverse wave will still excite it. It is quite obvious if you think for a second: most loudspeakers in the world are membranes, and an oscillating membrane is indistinguishable from a surface acoustic wave.
 
  • #16
The eardrum can be considered as a membrane and this responds most strongly to longitudinal vibrations. It is sound that's being discussed,not vibrations in general.
 
  • #17
Dadface said:
The eardrum can be considered as a membrane and this responds most strongly to longitudinal vibrations. It is sound that's being discussed,not vibrations in general.

What's your definition of "sound"? And why is your definition relevant for me or anyone else?
 
  • #18
Dadface said:
I think the answer to the OPs question is no. Sound waves are longitudinal and not transverse. By definition sound waves are related to those types of waves that can be sensed by the ear and the ear drum reacts to longitudinal vibrations.
A solid may be able to vibrate in different ways but only the longitudinal components of those vibrations can be transferred to any surrounding air as sound waves. It reminds me of the longitudinal and transverse vibrations that can be set up on a stretched string. Both set up longitudinal vibrations (sound waves) in the surrounding air.

apparently sound can be longitudinal and transverse wave but air and fluid cannot support transverse wave and sound can be transverse wave in solid
i need a book to comfirm the hypothesis
 
  • #19
voko said:
What's your definition of "sound"? And why is your definition relevant for me or anyone else?

Vibrations which travel through the air or another medium and are sensed by the ear.
(Concise Oxford English Dictionary)

The relevant part of this discussion is that the vibrations "are sensed by the ear" and the ear responds to longitudinal vibrations. Any transverse components eg along the plane of the ear drum will not be effectively passed through.
I'm not sure at what level of education the op is at but I have a reasonable idea. In the UK system any reference to sound being carried by transverse waves will lose marks.
 
  • #20
Dadface said:
Vibrations which travel through the air or another medium and are sensed by the ear.

So ultrasound is not sound then?

The relevant part of this discussion is that the vibrations "are sensed by the ear" and the ear responds to longitudinal vibrations.

I am pretty sure that if a solid bar is made to touch the eardrums, a surface acoustic wave on it will be registered.

The only reason you can talk about the ear responding to longitudinal vibrations is because ordinarily it senses vibrations in air, which does not support any other. So you are effectively reducing sound to a phenomenon occurring only in air (or perhaps other gases and liquids). Which is at odds with the widely known and recognized concept of sound in solids.
 
  • #21
This is yet another thread dealing with definition and classification. A lot of it is wasted effort because people are not talking from the same standpoint..
You can 'define' sound as the sort of vibrations that the ear can detect by longitudinal vibrations of the air in the ear or you can define it in terms of the frequency range. This is not really relevant to the modes of transmission of vibrations through media yet contributors are running round in circles trying to convince each other one way or another.
There is very little to be gained in the way of understanding, merely by putting things in columns of what they are and what they aren't. The UK has recently found itself lower in the education league tables. Not surprising, as kids are often taught their Sciences by teachers who stopped doing Physics, Chemistry or Biology after Double Award Science. Classification is favourite under those circumstances. How ye sow, so shall ye reap.
 
  • #22
That's a questions about semantics not physics. The answer to the question depends on what the definition of a sound wave is. Usually the longitudinal waves in a solid are called p-waves while the transverse waves in a solid are called s-waves. both of them can excite longitudinal waves in the air which will be perceived as sound.
 
  • #23
The effect at the surface will be due to a surface wave, unless the sound wave is normal incidence. Surface waves tend to be neglected in many situations particularly when Seismic waves are taught in school. But they are very relevant to buildings, which are on the surface and when sound energy is coupled into and out of a solid surface.
More over simplification due to classification, I think.
 
  • #24
voko said:
So ultrasound is not sound then?
I think it's well understood by all here that ultrasound is sound and we could go for more detailed definitions such as defining "sounds which are audible to humans" or to bats or to any other animal species. But I don't want to get involved in semantics but want to see OPs question answered at a suitable level.
I'm assuming that the OP is currently at a level which is equivelent to GCSE or perhaps AS or A level (UK qualifications).If so the specifications require that students should understand that....."Sound waves are longitudinal" (P1.5.1b AQA physics)
 
  • #25
So Ultraviolet is 'Light' and Infra Red is 'Light' too - but microwaves and X Rays aren't? Classification really gets in the way of things.
That quote from the AQA Specification just makes my point about classification taking over from understanding. It would have been just as easy to cover themselves and say that sound waves in air are longitudinal. Let's face it, the 'sound' that is transmitted along the ossicles, is in the form of lateral vibrations on levers. But I would be surprised if the person who approved the spec gave it as much thought as has been expended on this thread.
 
  • #26
AQA and all the other specs don't go into detailed classification in the way that is being suggested as should be evidenced from the quote.As far as EM waves are concerned the visible spectrum is considered to be that range of frequencies detectable by humans only. As far as ultrasound is concerned bats and dolphins have been covered by some syllabi.
The classification and definition required are,in my opinion, at the right level for the courses being studed.
 
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  • #27
I am not suggesting that the specs should go into classification. Quite the reverse, (in fact have you read what I have written about classification?). I say that they should just state things clearly and unambiguously and give adequate, watertight definitions. If there is possible doubt, then things should be made clearer. How can you say that there is a "right level" for the course when the information is, in fact, incorrect or inadequate.? Which level of student are your remarks aimed at? The GCSE course is supposed to be aimed at all levels. The modern specifications are so inconsistent. On the same page, they refer to the most sophisticated of modern Science concepts and also use the sloppiest terms to discuss the very basic fundamentals of Science.
That reptile Gove has said so many things about British education that are perfectly accurate and people who support the present, flawed system are just giving him more excuses for his barbaric plans for change. Perhaps you are one of his fans?

You cannot predict how a bright student will receive a piece of badly stated information. It will be the bright student who sees the holes in inadequate teaching and will suffer because of them. The specifications are written, largely with a view to the lowest common denominator of the (non-specialist) staff who are required to deliver the stuff and not to the highest ability students.
These forums are littered with questions from students who have been confused because the information (the trite and over simplified stuff) they are given, has just not been thought through properly. I could point to dozens of posts which show how a poor syllabus has harmed the progress of high flyers.
 
  • #28
Dadface said:
I'm assuming that the OP is currently at a level which is equivelent to GCSE or perhaps AS or A level (UK qualifications).If so the specifications require that students should understand that....."Sound waves are longitudinal" (P1.5.1b AQA physics)

I can't be bothered to read what the AQA says this week (and they will probably say something different next week), but as sophiecentaur said, the bald statement "Sound waves are longitudina" will satisfy the kid with no interest in science who wants to scrape a C grade (and it will also satisfy teachers with the same objectives!) - but if a bright kid actually plucks a guitar string (or even twangs a ruler on a desk) and observes what happens, he or she is likely to get confused!
 
  • #29
AlephZero said:
I can't be bothered to read what the AQA says this week (and they will probably say something different next week), but as sophiecentaur said, the bald statement "Sound waves are longitudina" will satisfy the kid with no interest in science who wants to scrape a C grade (and it will also satisfy teachers with the same objectives!) - but if a bright kid actually plucks a guitar string (or even twangs a ruler on a desk) and observes what happens, he or she is likely to get confused![/QUOTE

Where are you getting this information from? In AS specs kids study SHM and waves in strings in considerable detail.They know that both transverse as well as longitudinal vibrations of the string the ruler, or whatever it is sets up longitudinal waves in the surroundings which, depending on the frequency, can be detected as sound. I rarely see anyone get confused by that. Have you actually met any teachers who have the objectives you referred to?
 
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  • #30
sophiecentaur said:
I am not suggesting that the specs should go into classification. Quite the reverse, (in fact have you read what I have written about classification?). I say that they should just state things clearly and unambiguously and give adequate, watertight definitions. If there is possible doubt, then things should be made clearer. How can you say that there is a "right level" for the course when the information is, in fact, incorrect or inadequate.? Which level of student are your remarks aimed at? The GCSE course is supposed to be aimed at all levels. The modern specifications are so inconsistent. On the same page, they refer to the most sophisticated of modern Science concepts and also use the sloppiest terms to discuss the very basic fundamentals of Science.
That reptile Gove has said so many things about British education that are perfectly accurate and people who support the present, flawed system are just giving him more excuses for his barbaric plans for change. Perhaps you are one of his fans?

You cannot predict how a bright student will receive a piece of badly stated information. It will be the bright student who sees the holes in inadequate teaching and will suffer because of them. The specifications are written, largely with a view to the lowest common denominator of the (non-specialist) staff who are required to deliver the stuff and not to the highest ability students.
These forums are littered with questions from students who have been confused because the information (the trite and over simplified stuff) they are given, has just not been thought through properly. I could point to dozens of posts which show how a poor syllabus has harmed the progress of high flyers.

Please show me examples of where you think the syllabi are lacking.
 
  • #31
Dadface said:
AlephZero said:
I can't be bothered to read what the AQA says this week (and they will probably say something different next week), but as sophiecentaur said, the bald statement "Sound waves are longitudina" will satisfy the kid with no interest in science who wants to scrape a C grade (and it will also satisfy teachers with the same objectives!) - but if a bright kid actually plucks a guitar string (or even twangs a ruler on a desk) and observes what happens, he or she is likely to get confused![/QUOTE

Where are you getting this information from? In AS specs kids study SHM and waves in strings in considerable detail.They know that both transverse as well as longitudinal vibrations of the string the ruler, or whatever it is sets up longitudinal waves in the surroundings which, depending on the frequency, can be detected as sound. I rarely see anyone get confused by that. Have you actually met any teachers who have the objectives you alluded to?

Well, if he hasn't, i certainly have. Plenty of teachers enter secondary Science education with nothing more than a GCSE Double award C grade in Physics and Chemistry plus, perhaps a degree in Biology. They are delighted at every student who gets C and above and very often, wouldn't spot a potential First Class Hons in Physics because they just don't know enough. It's not their fault; it's the system and they need a job. I never met one of them who actually wanted to be delivering Physics.

At AS, quite recently, kids were hit, first of all, with Fundamental Particles - involving the classification of particles in terms of Quarks. This was before they even knew the definition of Momentum or what an electron Volt was. A shameless bit of 'bums on seats' if ever there was one. Can you justify that?
 
  • #32
Dadface said:
"Sound waves are longitudinal" (P1.5.1b AQA physics)

I have no idea what kind of physics this "AQA Physics" is, but it is definitely not the physics that applies to my universe. In my universe, sound propagates in solids just as well as it does in fluids, and propagation in solids involves transverse waves.

The argument "let's not confuse the uneducated inquirer and let's feed him some half-truth and stick with it", even when the inquirer has specifically asked for the other half of the truth, would be received well in the Middle Ages, but now it feels distinctly dated.
 
  • #33
Dadface said:
I think it's well understood by all here that ultrasound is sound and we could go for more detailed definitions such as defining "sounds which are audible to humans" or to bats or to any other animal species. But I don't want to get involved in semantics but want to see OPs question answered at a suitable level.
I'm assuming that the OP is currently at a level which is equivelent to GCSE or perhaps AS or A level (UK qualifications).If so the specifications require that students should understand that....."Sound waves are longitudinal" (P1.5.1b AQA physics)

Can you translate those qualifications into something that makes sense?
 
  • #34
Dadface said:
I think it's well understood by all here that ultrasound is sound and we could go for more detailed definitions such as defining "sounds which are audible to humans" or to bats or to any other animal species. But I don't want to get involved in semantics but want to see OPs question answered at a suitable level.
I'm assuming that the OP is currently at a level which is equivelent to GCSE or perhaps AS or A level (UK qualifications).If so the specifications require that students should understand that....."Sound waves are longitudinal" (P1.5.1b AQA physics)

A level
i just want to know the truth not what books want me to know

so is sound wave a transverse wave in solid?
 
  • #35
Three modes of transmission are possible in a solid - longitudinal, transverse and surface. It depends upon how you launch the wave as to what you will get. The only thing to remember is that fluids do not support transverse vibrational waves.
 
  • #36
It should also be added that sound propagation in solids is audible directly. There is a mechanism known as "bone conduction", where sound propagates through the scull bones directly into the inner ear. This mechanism is responsible for the surprise when we hear our own voices recorded: they appear higher than we are accustomed to. This is because the bone conducts the lower tones better than air.

Bone conduction most definitely uses both longitudinal and transverse waves.

This mechanism is employed in hearing aids and some special equipment such as underwater communications. It was recently used with Google Glass.

So sound is most definitely not longitudinal waves only, no matter what some dumbed down teaching standards would want us to believe.
 
  • #37
jerry0696 said:
A level
i just want to know the truth not what books want me to know

so is sound wave a transverse wave in solid?

In a solid you may have both transverse sound waves (called s-waves) and longitudinal sound waves (called p-waves). On a (Newtonian) liquid or gas you only get longitudinal sound waves. Non-Newtonian fluids may behave more like solids at times and might sustain s-waves as well.
 
  • #38
dauto said:
In a solid you may have both transverse sound waves (called s-waves) and longitudinal sound waves (called p-waves). On a (Newtonian) liquid or gas you only get longitudinal sound waves. Non-Newtonian fluids may behave more like solids at times and might sustain s-waves as well.

thank you
 
  • #39
sophiecentaur said:
Dadface said:
Well, if he hasn't, i certainly have. Plenty of teachers enter secondary Science education with nothing more than a GCSE Double award C grade in Physics and Chemistry plus, perhaps a degree in Biology. They are delighted at every student who gets C and above and very often, wouldn't spot a potential First Class Hons in Physics because they just don't know enough. It's not their fault; it's the system and they need a job. I never met one of them who actually wanted to be delivering Physics.

At AS, quite recently, kids were hit, first of all, with Fundamental Particles - involving the classification of particles in terms of Quarks. This was before they even knew the definition of Momentum or what an electron Volt was. A shameless bit of 'bums on seats' if ever there was one. Can you justify that?

Perhaps you have not met many teachers. The ones I have known can spot potential and would certainly not be delighted if a student who they knew to be capable of getting an A star ended up getting a C.
I don't understand the points you made in your second paragraph. All the schools and FE colleges I know require that students have a good pass at GCSE Physics before they are allowed on to an AS course.They would already have a good working knowledge of momentum because they would have studied it at GCSE. Of course a good teacher would do a recap of the subject at an appropriate time during the AS studies. That recap and the introduction of the eV would be covered at appropriate points during the studies.
 
  • #40
voko said:
I have no idea what kind of physics this "AQA Physics" is, but it is definitely not the physics that applies to my universe. In my universe, sound propagates in solids just as well as it does in fluids, and propagation in solids involves transverse waves.

The argument "let's not confuse the uneducated inquirer and let's feed him some half-truth and stick with it", even when the inquirer has specifically asked for the other half of the truth, would be received well in the Middle Ages, but now it feels distinctly dated.

Bearing in mind that one should consider the the present educational level of any enquirer when answering a question it is a universe shared by many educational establishments organisations and books most being very prestigious.A quick search threw up the following, on just the first page
1.Salford Universities learning resource for GCSE describe sound waves as being longitudinal
2.Indiana university refer to it as a pressure wave.
3.Canadas Science and Technology Museum described sound as being a vibration or wave of air molecules
4.The Glen Research Centre of NASA describe sound as being a sensation created in the human brain in response to small pressure fluctuations in the air.

Of course one should try to provide any enquirer with additional information asked for. The fact that there are different modes of vibration in solids has never been disputed here.
 
  • #41
dauto said:
Can you translate those qualifications into something that makes sense?

GCSE stands for general certificate of education. Students study different subjects at GCSE and are examined, the finals usually being when they are fifteen or sixteen.
Students who continue their studies usually specialise in a smaller number of subject four being the average. AS stands for advanced supplementary and is a one year course.A stands for advanced level and is an additional year after AS.
Amongst other things most universities require three good passes at A level.
 
  • #42
jerry0696 said:
but is sound transverse ,longitudinal or both in solid??
everyone is telling me that sound can only be longitudinal (my teachers)

your teachers tell you that because that's what's in your curriculum (same here ) so I suggest that you stick to what your teachers say in exams and scolastic stuff but know and understand the truth
 
  • #43
voko said:
It should also be added that sound propagation in solids is audible directly. There is a mechanism known as "bone conduction", where sound propagates through the scull bones directly into the inner ear. This mechanism is responsible for the surprise when we hear our own voices recorded: they appear higher than we are accustomed to. This is because the bone conducts the lower tones better than air.

Bone conduction most definitely uses both longitudinal and transverse waves.

This mechanism is employed in hearing aids and some special equipment such as underwater communications. It was recently used with Google Glass.

So sound is most definitely not longitudinal waves only, no matter what some dumbed down teaching standards would want us to believe.

It certainly sounds faesible that bone conduction involves different modes of vibration and surely transverse vibrations must be included. Take care though when assuming too much. I'm referring to the work on hearing through bones being carried out by Professor Puria at Stanford University.
The team found that the application of inertial bone stimulation only resulted in hearing loss but when bone compression was included hearing jumped up. I think this is work in progress.

And why so scathing about work being dumbed down as you describe it? Simplifying to a relevant level is one of the ways education can be delivered effectively. As for this discussion I think that the syllabi cover sound studies at a proper and useful level. Students should be informed when things are simplified so they have the opportunity of looking at stuff in greater depth.
 
  • #44
kira506 said:
your teachers tell you that because that's what's in your curriculum (same here ) so I suggest that you stick to what your teachers say in exams and scolastic stuff but know and understand the truth

Good advice.
 
  • #45
Dadface said:
Good advice.

Pragmatic I agree but what do you say to the student who is offended that he was told the 'wrong' stuff, earlier on? They learn to be suspicious of everything they are ever told. Can't be a good thing for their confidence.
Just a bit of thought in how you tell them things can prevent that sort of thing happening. There are plenty of suitable ways of describing the propagation of sound which do not lead to the OP's problem. Unfortunately, specifications do not seem to include such subtleties. Doesn't the evidence of low UK performance in international league tables support my concern?

btw, unless things have changed greatly in the last year or two, it is quite possible for kids to find themselves on an AS Physics course with no more than a reasonable pass in GCSE Double Award. So they may never (and I mean never) have been taught by a Physicist before their AS course. Momentum is just one of the things that the Double Award leaves out. I know there are many 'good' schools where such problems don't arise but taking the three Sciences separately usually seems to require extra curricular time and more work for the Science staff.
 
  • #46
Dadface said:
And why so scathing about work being dumbed down as you describe it? Simplifying to a relevant level is one of the ways education can be delivered effectively. As for this discussion I think that the syllabi cover sound studies at a proper and useful level. Students should be informed when things are simplified so they have the opportunity of looking at stuff in greater depth.

This is not just simplifying, this is over-simplifying to the level when it is just wrong. "Audible by humans" is wrong. "Longitudinal only" is wrong. This is not a simplification, this is misinformation.
 
  • #47
sophiecentaur said:
Pragmatic I agree but what do you say to the student who is offended that he was told the 'wrong' stuff, earlier on? They learn to be suspicious of everything they are ever told. Can't be a good thing for their confidence.
Just a bit of thought in how you tell them things can prevent that sort of thing happening. There are plenty of suitable ways of describing the propagation of sound which do not lead to the OP's problem. Unfortunately, specifications do not seem to include such subtleties. Doesn't the evidence of low UK performance in international league tables support my concern?

btw, unless things have changed greatly in the last year or two, it is quite possible for kids to find themselves on an AS Physics course with no more than a reasonable pass in GCSE Double Award. So they may never (and I mean never) have been taught by a Physicist before their AS course. Momentum is just one of the things that the Double Award leaves out. I know there are many 'good' schools where such problems don't arise but taking the three Sciences separately usually seems to require extra curricular time and more work for the Science staff.

I think the main problem, as you pointed out earlier is the shortage of well qualified staff, including those with enough experience to guide new colleagues.
 
  • #48
voko said:
This is not just simplifying, this is over-simplifying to the level when it is just wrong. "Audible by humans" is wrong. "Longitudinal only" is wrong. This is not a simplification, this is misinformation.

I have a suggestion. Look at the syllabi of different exam boards and other relevant organisations and suggest to them that they tweak their specs. For example part of an AQA spec states:

...sound waves are longitudinal...

This could be added to slightly, for example:

...sound waves are longitudinal but candidates should also be aware that in solids sound waves can be transverse as well as longitudinal...(I'm sure the wording can be improved)

If you give it a go please let us know how you get on.
 
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  • #49
Dadface said:
I think the main problem, as you pointed out earlier is the shortage of well qualified staff, including those with enough experience to guide new colleagues.

There is an excuse, of sorts, for the lack of qualified staff but there is no excuse for poor specifications and for education policies that build in unnecessary problems.
If the UK want good teachers then they just need to pay enough to make the profession attractive to enough bright people and for the Great British Public to realize that Education is something to aspire to. The recent surge in better quality graduates, due to the lack of other jobs, may produce an improvement in the medium term but, once the better jobs become available elsewhere, we will be back to an intake of teachers for whom the profession has been largely a second choice.
The present government doesn't seem to know what it means when it talks about 'qualified' or 'unqualified' teachers, for a start. We don't need any David Starkeys in charge of classes in our schools. (ref. that disastrous TV prog).
 
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