# Confused with mechanical waves

#### Woopydalan

Hello,

I am having issues understanding all the waves, there are a lot of equations involving sine and cosine..and then all the various ways frequencies are looked at. I'm talking about pendulums, oscillators, standing waves, and sound waves. They all look similar in there form, but it is a large discombobulation of equations involving the trig functions

Any way to consolidate all this information?

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#### Simon Bridge

Homework Helper
They are all related to each other ... but you basically have to get used to it.

Pay attention to "simple harmonic motion" - all other periodic motion can be expressed as a sum of SHM.

The general equation for oscillating motion is the driven-damped harmonic equation (look it up). What you see are specific solutions for common situations.

Lastly - you will find out about the wave equation.

Mostly you just have t wait - your course is taking you through the simple solutions first to get you used to the ideas ... as you advance through your course, you will find things start to consolidate.

#### Studiot

Simon is right but here is one comment that may help.

An oscillation or vibration is what happens to a single object eg a pendulum. It is not a wave.

What happens is periodic that is the motion repeats itself regularly in a set time interval. eg the pendulum swings backwards and forwards.

Note that periodic motion that does not go backwards and forwards over its path, but repeats its path in one direction only, is not an oscillation. An example is circular motion.

When there are lots of individual bodies (oscillators) that are all mechanically linked in some way the motion of one can be transmitted down the chain of linked oscillators.

This is wave motion.

A good example is a sound wave where small parcels of air alternately expand and contract.
As the first parcel expands it pushes or squashes the second one thus transmitting its motion to the second one and so on down the line of parcels. This handing on the expansion/contraction down the chain forms the sound wave and each parcel forms an individual oscillator.

The wave motion is a separate motion from the motion of each individual oscillator.
It is a property of the collection of oscillators as a whole.
The oscillation is a property of each individual body or oscillator.

We call the simplest and most common oscillation simple harmonic motion.

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#### Simon Bridge

Homework Helper
A good example is a sound wave where small parcels of air alternately expand and contract.
Air particles expand and contract? I thought the particles bunched up and um down? Y'know - pressure waves?

Nice point about the difference between waves and other kinds of periodic motion.
Some confusion can come from using the word "wave" to describe the wave-shape[/i of a function and the wave properties.

#### Studiot

Air particles expand and contract? I thought the particles bunched up and um down? Y'know - pressure waves?
This is a very simplified explanation.

If a parcel (volume) of gas expands its pressure decreases and if it contract its pressure increases.

Voila a pressure oscillation in that particle.

#### Simon Bridge

Homework Helper
Actually I've seen a description like, only for boiling, in a chef-course text book ... so I did kinda realize ;) Need to hear from OP.

#### Woopydalan

ok so why is it that the waves have similar equations with simple harmonic motion, even though they are two separate things?

#### Studiot

I'm sorry, where did I mention equations?

#### Woopydalan

So in essence, a wave is multiple oscillations occurring simultaneously?

#### Studiot

Yes, but not at the same location.

#### Woopydalan

Ok now for my next question, the equations describing oscillations and waves are very similar, but have some differences.

How do the equations account for these differences?

#### Studiot

Ok now for my next question, the equations describing oscillations and waves are very similar, but have some differences.
Actually they are very different.

Look closely

SHM y = asin(ωt)

WM y = asin(x-ct)

The first is an equation between displacement, y, and time, t. Nothing else - no other variables involved.

The second is an equation between displacement, y, and another distance, (x -ct). The constant c (a velocity) multiplied by time keeps the units consistent.

So SHM is independent of x, wave motion is not.

#### Simon Bridge

Homework Helper
y = asin(x-ct)
... for a travelling wave with unit wave-number or distance measured in units of the wavelength over two-pi.

For a standing wave y(x,t)=A[sin(x-ct)+sin(x+ct)] I guess.

We could illustrate the similarity along with the difference by observing that SHM is y(t)=Asin(ωt) while a travelling wave has a space component: y(x,t)=Asin(kx-ωt).

The similarity between shm and wave motion comes from how every point x has something executing SHM. eg. at x=0, y(x,t)=Asin(-ct) for Studiot's example. You can think of wave-motion as being the result of coupled SHM oscillators ... where the motion of one depends partly on the motion of others close by.

The math looks like that because the number inside the brackets of a trig function has to be an angle ... something like time has the wrong units - so we cannot just leave it there. ω=2π/T (where T is the time for one oscillation) means that ωt = θ with units of radiens. (If you want it in degrees then ω=360/T)

A simple example would be a row of masses on a frictionless surface joined together by springs. They are all "mass on a spring" oscillators - so get a term that looks kinda like SHM. But they are attached to each other too - so they can transfer energy between masses.

Another example is to go to a playground where there are swing sets - with swings in a row. The kind with chains are best. You can set each swing going and they don't affect one another ... by themselves they are executing (damped+driven) harmonic motion. The motion is no longer SHM but that's real life for you - a good swing set should be close enough for a while for our purposes.

To see wave motion, tie adjacent chains together about 1/3 down from the top, when all the swings are still. The set one going only. Watch.

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#### Woopydalan

Ok, now some other confusion comes with keeping all the λ_n straight. Do the multiples of wavelengths come in when boundary conditions are set for the wave. When the air is oscillating in a tube, is the boundary condition the tube that is limited how high the amplitude of the wave can be, or maybe that is irrelevant since sound is a longitudinal wave?

How do I keep all the frequencies involved straight?

#### Studiot

Ok, now some other confusion comes with keeping all the λ_n straight.
Yes it certainly does. What are the λ_n and where did they come from?

#### Woopydalan

Ok so the two situations involving n multiples of λ are boundary conditions. Usually we deal with strings vibratings attached to two walls, hence boundary conditions. I am thinking that the waves must have discrete λ's because the walls are such that both ends must be nodes, so the wave needs to contour itself to be able to fit into the walls.

With sound waves, the boundary conditions are the pipes with either one end closed or both ends open. With one end closed and one end open, the end must be a node (pressure node right?) as the boundary forces the wave to reflect back up. The opening is an antinode.

Once again the wave must contour itself to have a node at the closed end and antinodes at the open ends.

Thus quantization of frequencies?

Am I missing anything here? I have issues solving problems where it will ask me to find all the antinodes within the boundary conditions and things of that nature.

#### Redbelly98

Staff Emeritus
Homework Helper
Ok so the two situations involving n multiples of λ are boundary conditions. Usually we deal with strings vibratings attached to two walls, hence boundary conditions. I am thinking that the waves must have discrete λ's because the walls are such that both ends must be nodes, so the wave needs to contour itself to be able to fit into the walls.

With sound waves, the boundary conditions are the pipes with either one end closed or both ends open. With one end closed and one end open, the end must be a node (pressure node right?) as the boundary forces the wave to reflect back up. The opening is an antinode.

Once again the wave must contour itself to have a node at the closed end and antinodes at the open ends.
Yes, in the types of problems you mention, the boundary conditions determine which wavelengths are physically possible. In introductory physics problems, the usual boundary condition is that there be a node -- or sometimes an antinode -- at each end of the pipe or string.

It sounds like you have a reasonable understanding of the situation.

Thus quantization of frequencies?
Yes. Quantization of wavelengths implies quantization of frequencies (since the wave speed is constant, and f = v/λ)

Am I missing anything here? I have issues solving problems where it will ask me to find all the antinodes within the boundary conditions and things of that nature.
Well, you know that each end is a node or possibly an antinode. The nodes occur every half-wavelength, as do the antinodes. And the nodes and antinodes are a quarter-wavelength from each other. I'm not sure where the difficulty is -- once you have figured out what the wavelength is.

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#### Woopydalan

Maybe its just a mesh of putting all these ideas together. Often the problems have some small twists to them, and I get confused putting the pieces together to solve the problem.

#### Simon Bridge

Homework Helper
I am thinking that the waves must have discrete λ's because the walls are such that both ends must be nodes, so the wave needs to contour itself to be able to fit into the walls.
You are thinking of standing waves in different conditions - yes, they come from the boundary conditions when solving the wave equation. eg.
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/wavsol.html

When you vibrate a string (say), you get two travelling waves going in opposite directions and reflecting from the boundaries. The waves will destructively interfere and you get nothing or a ind of mess, except at particular speeds, where they reinforce to give the orderly harmonics you have seen.

You get nodes at the points of maximum destructive interference of the two waves - this will be automatic for the fixed ends of a string because the wave inverts on reflection there.

If the end is not fixed, though, the wave does not invert at reflection and you get an antinode there.

It is possible to have a situation where the string in not fixed, but not completely free to move ... and you get less well defined behavior. It is probably a good idea to get used to the simple situations before tackling that.

Maybe its just a mesh of putting all these ideas together. Often the problems have some small twists to them, and I get confused putting the pieces together to solve the problem.
A theme throughout has been that you have been thrown by the similarity in the math for different situations ... this can happen when you have been used to a problem-solving method that requires that you "find the right equation" and then plug numbers into it.

You have to move away from that - let the physics dictate the form of the equation, then the math becomes a language.
Keep the physics in mind and these confusions go away.... and you don't have to memorize so many almost-alike equations ;)

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#### Woopydalan

I don't know why in simple harmonic motion the displacement x is not a factor, whereas in wave motion it is. Is this because with SHM, the spring is translating only back and forth, so it is moving in one direction, so you can't have two types of locations in space affecting the equation?

With the string vibrating, the wave is propagating forward, yet the elements on the string go up and down, so there are two components of location involved, hence two location parameters involved in the equation? the x referring to where the element is on the string and the y to mean where the wave is at that particular element (does the Y imply its vertical displacement?)

With sound waves, the air particles are moving longitudinally, essentially vibrating back and forth, so I don't see why you can have two parameters of location involved, since there is no vertical displacement.

#### Simon Bridge

Homework Helper
Like for the horizontal mass-on-a-spring you mean ... in which case x(t)=x(0)+Asin(ωt+δ) ... where x(0) is the x location of the equilibrium position.

Normally you'd put y(t)=x(t)-x(0)

That would be SHM. Is it a solution to the wave equation?

Remember the variable y can be anything - it does not have to be a transverse displacement.

For the sound wave ... x(t) itself depends on x... which can get confusing:
We'd define y(t) the same way and write y(t)=Asin(kx-ωt+δ)
(IRL we describe sound waves in terms of air pressure.)

Now - what was your question?

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#### Woopydalan

Ok so first you said to think of the physics and forget the math, then when I try to understand the physics you use the math as your safety net of explanation :P

For sound waves

S(x,t) = S_max cos(kx - ωt) (Why is the phase angle omitted??)

Imagining a piston oscillating back and forth, it pushes air molecules toward the end of the tube, when it pushes the batch of air, there is less air density at that area when the piston returns back to its initial position, so the air molecules return back to fill that void, hence oscillations of the batches of air. The original air pushes another batch of air, then the air ensity in that spot is less so the 2nd batch returns, and that behavior continues.

This situation is what s is referring to, yes? Such that the S_max is the amplitude of the oscillations of the individual batches of air. The x is refering to where along the line in space that batch of air molecules is present? And S(x,t) is refering to where in that individual range that the air molecule batch can oscillate? So it makes sense that the batches translate and oscillate on the same axis, so S(x,t) is the position along that x axis. I think this makes sense to me, is it right? I can see why the position x is important in determing where a certain batch of air located on the x axis is at a certain time in its oscillation

#### Woopydalan

Also, why is it that sinusoidal waves are used to represent sound, when it is longituinal and not transverse?

Is the difference between a standing wave and a non-standing wave boundary conditions?

#### Simon Bridge

Homework Helper
Ok so first you said to think of the physics and forget the math, then when I try to understand the physics you use the math as your safety net of explanation :P
I probably also said to use the math as a language ;)
I suspected that you may have been taking the use if y to literally mean a transverse displacement and I wanted to show you that it can mean anything I want.

It seems what I actually said was:
me said:
you have been used to a problem-solving method that requires that you "find the right equation" and then plug numbers into it.
You have to move away from that - let the physics dictate the form of the equation, then the math becomes a language.
Keep the physics in mind and these confusions go away....
... moving away from a "find the right equation" approach to problem solving is not the same as "forget the math".

For sound waves

$S(x,t) = S_{max} \cos(kx - \omega t)$ (Why is the phase angle omitted??)
It isn't - the relative phase angle is 0 for this particular example if the general equation is y(x,t)=Acos(kx-ωt+δ) and π/2 if the general equation is y(x,t)=Asin(kx-ωt+δ).

The phase angle, or just "phase" for short, is θ=kx-ωt+δ ... the δ is the relative phase for the situation that S(0,0)≠Smax.

Imagining a piston oscillating back and forth, it pushes air molecules toward the end of the tube, when it pushes the batch of air, there is less air density at that area when the piston returns back to its initial position, so the air molecules return back to fill that void, hence oscillations of the batches of air. The original air pushes another batch of air, then the air ensity in that spot is less so the 2nd batch returns, and that behavior continues.

This situation is what s is referring to, yes?
There are lots of ways to make an oscillating pressure wave and that would be one of them, yes.
Such that the S_max is the amplitude of the oscillations of the individual batches of air. The x is refering to where along the line in space that batch of air molecules is present?
For a particular molecule, x is it's equilibrium position
And S(x,t) is refering to where in that individual range that the air molecule batch can oscillate?
It tells you how far from it's equilibrium position a molecule normally at position x has been shifted.

Easier to understand if S is pressure ... a pressure gauge at position x will show a reading of S(x,t).

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#### Woopydalan

Was my interpretation of s(x,t) incorrect? I'm thinking the equation must be describing where a particular batch of air is at a certain time.

So x is the equilibrium position of the batch of air, so once it is disturbed and is displaced, it is no longer in equilibrium. Knowing the equilibrium position x and its displacement s(x,t) on its oscillation, we can determine where that particular batch of air is at any given time?

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