# Wavelength and amplitude. Are they related?

1. Aug 19, 2008

### sameeralord

Hello everyone,

I just want to ask a question. If higher the amplitude the particles move a larger distance right. So is the wavelength related to amplitude. The sound travels in circles right. So I'm bit confused here how they travel in circles and be a longtitudinal wave at the same time. Any help would be appreciated. Thanks

2. Aug 19, 2008

### bassplayer142

3. Aug 20, 2008

### Staff: Mentor

Sinusoidal displacement, not circles. Check out the link bassplayer provided.

4. Aug 20, 2008

### sameeralord

How can sound be sinusodial when it is not a transverse wave.

I'm referring to this. Isn't sound kind of circular. Is anyone able to tell me how this wave relates with the longitudinal wave. For example is the wavelength here the distance between two waves.

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5. Aug 20, 2008

### Topher925

Sound doesn't manifest itself in two dimensions, its does it in 3. Consider a bomb exploding, giveing off a shock wave in every direction. You can think of this as a sound pulse or a piece of a sound wave. Sound itself is an expansion and compression of molecules that propagates through a median. Don't think of sound as particles moving up and down or left and right, but getting squished together and then pulled apart in a spherical pattern.

6. Aug 20, 2008

### bassplayer142

Exactly ^. Think of the sinisoidal wave as a way of representation of the sound wave. The crest of the wave would be the point of high density and the trough would be the point of low density. The middle would be no change or would represent a room with no noise.

http://science.hq.nasa.gov/kids/imagers/ems/wave_crest.gif

7. Aug 20, 2008

### Staff: Mentor

Choose a point somewhere in that diagram and imagine an air molecule at that location. Imagine a straight line that extends radially outward from the source, through the molecule. As the sound wave passes, the molecule oscillates sinusoidally, back and forth along that line, with the center point of the oscillation remaining fixed.

This page has a nice animation of molecules in a sound wave. It's a plane wave, not a spherical wave, but the basic idea is the same. Watch individual molecules carefully and see how they each oscillate. The phase of the oscillation varies with location, so the net effect is a "density wave" that moves from left to right.

Last edited: Aug 20, 2008
8. Aug 22, 2008

### atyy

Wow, good question. I never thought about that. It does seem like a logical necessity for a longitudinal wave, whereas it doesn't necessarily have to happen for a transverse wave.

Of course we don't trust wikipedia:
http://en.wikipedia.org/wiki/Nonlinear_acoustics#cite_ref-0

I wonder if this is related:
http://www.woodynorris.com/Articles/TechnologyReview.htm

9. Aug 22, 2008

### Staff: Mentor

No, it's not. Please see the previous discussion.

10. Aug 22, 2008

### atyy

The previous discussion only seems to work for small displacements.

11. Aug 22, 2008

### Staff: Mentor

Why do you say that? There are no non-linearities involved.

12. Aug 22, 2008

### FredGarvin

That Woody Norris site is boarderline with me. I doubted it the first time it showed up. You notice it hasn't been updated since 2006? I wouldn't waste my time.

How was the assumption about small displacements arrived at?

13. Aug 22, 2008

### billiards

You're right, sound is kind of circular.

What you need to distinguish is the ray (the linear thing) from the wave front (the circular thing).

A ray is basically a line, that energy tends to follow.

A wavefront is an envelope the encapsulates all of the rays.

14. Aug 22, 2008

### atyy

I got this from Google books:
"Before discussing non-linearity arising from the physics of particular systems, we consider a purely kinematic effect which is present in all longitudinal waves." (Iain G. Main, Vibrations and Waves in Physics, 3 ed, 1993, page 288).

I don't know the details, having never thought about it until sameeralord's question. I think he's a bit confused about the circular part, but he makes a good point that longitudinal waves are necessarily nonlinear at large amplitudes, whereas transverse waves can conceivably be linear even at large amplitudes.

An electromagnetic wave in vacuum as described by Maxwell provides an example of a transverse wave that is linear at large amplitudes, while it's very had to find longitudinal waves that are linear at large amplitudes. However, it is actually wrong to appeal to experiment on this point, since the claim is "logically necessary".

15. Aug 23, 2008

### billiards

Okay, let me try answer these points raised in the original post.

1. Yes, I would agree with this.

2. No! At least not in principle. However, there may be some confusion here: sound with shorter wavelengths (and thus higher frequencies) will tend to lose energy (attenuate) faster than sound with long wavelength (low frequency), so you might expect that by the time the sound had reached a distant observer it would be less trebly

3. The very crest of the first arriving energy is the wavefront. The wavefront of body waves, of which sound is a longitudinal example, will expand spherically in a homogeneous, iostropic medium, these are spherical waves, they require a half space. The example of the water wave is not strictly analagous to sound. The ripples on the surface of water expand cylindrically, these are cylindrical, surface waves.

Sound actually travels in more or less straight lines according to Fermat's principle, it take both the shortest and longest paths between two points, these are called rays.

4. I think your confusion here is mostly to do with not appreciating the difference between a wave front and a ray.

16. Aug 23, 2008

### atyy

I'm taking billiard's cue and organizing my reply to sameeralord's question in parts.

1. Yes.

2. Yes. Longitudinal waves are necessarily non-sinusoidal (ie. not the solution of a linear differential equation) at large amplitudes because the displacement of a particle in the medium is in the direction of energy propagation. This is in contrast to transverse waves which can be sinusoidal even at large amplitudes. In most everyday situations, eg. when you blast your stereo at full volume, the amplitude of a sound wave is still small enough that it can be approximated as a sinusoid. This was a great question you asked, I hadn't appreciated this before. (For a reference, see Iain G. Main, Vibrations and Waves in Physics, 3 ed, 1993, page 288).

3. If you drop a pebble in water, you will see that the ripples form a circular pattern around the pebble. The circular pattern expands in radius with increasing time, indicating that energy is travelling outwards from where you dropped your pebble. The energy is not travelling in circles. The energy is travelling outwards from where you dropped your pebble in straight lines. This is a perhaps more intuitive situation in which you see circles, but the energy travels in straight lines. The situation in the figure you showed is analogous, except that the pebble is replaced by a hi-fi speaker, the water is replaced by air, and the transverse wave in an incompressible medium is replaced by a (mostly) longitudinal wave in a compressible medium.

17. Aug 24, 2008

### schroder

1. If higher the amplitude the particles move a larger distance right.

2. So is the wavelength related to amplitude.

3. The sound travels in circles right.

4. So I'm bit confused here how they travel in circles and be a longtitudinal wave at the same time.

I guess I will follow the established format

1. Yes. The particles have a greater displacement but that displacement is limited within the wavelength of the wave itself.

2. No. The wavelength is related to the frequency of the vibration which is causing the sound wave.

At first, there might seem to be a contradiction between 1 and 2 above but there is none. The individual particles do have a greater displacement with greater amplitude, but that does not represent the wave displacement (wavelength). The particle displacement changes with the density or pressure within the wave but the particles cannot move any more than 1/4 wavelength. That is where the non-linearity comes into play in large amplitude (pressure) waves. The particles can be only squished together so much within the space of 1/4 wavelength so that any increase in force does not squish them any more. That is non-linearity. If the wavelength was related to the amplitude, then increasing the amplitude would increase the wavelength and reduce the frequency of vibration. That is not the case!

3. The waves are isotropically spherical but the sound radiates outward as longitudinal pressure waves.

4. See 3.

18. Aug 24, 2008

### atyy

I think the quote from schroder is correct. I just want to clarify how that and my earlier posts match up.

There are two ways of defining wavelength:
1) The minimum distance at which the entire waveform repeats itself. In this case, a square wave and a sine wave can have the same wavelength. This is the common sense definition, and it is a good one. However, this is not the sense in which amplitude is related to wavelength in a nonlinear wave.

2) There is a second definition of wavelength which says only the wavelength of sinusoidal waves is defined (in the same way that we can say that only the frequency of sinusoidal vibrations is defined - and hence we say ideal white noise contains high frequency components, although in terms of the minimum time at which the vibration repeats is infinite and hence low frequency). Since sameeralord was asking how sound can be sinusoidal, we should also consider this second definition. In this definition, nonlinearity is means that the waveform is not sinusoidal - and the amplitude is related to wavelength in the sense that at high amplitude more than one sinusoidal wavelength is present (eg. a square wave can be thought of as the sum of many sinusoidal waves).

Now, it should not be thought that a non-sinusoidal waveform implies nonlinearity - since a non-sinusoidal waveform can be made up of many sinusoidal waveforms. The nonlinearity is that the underlying differential equation predicts that a sinusoidal waveform is impossible, no matter how cleverly you displace the medium initially.

19. Aug 25, 2008

### atyy

This definition is not absolutely correct, but it is a good rule of thumb. Just for correctness, let me state that a nonlinear differential equation is one for which the sum of two solutions is not necessarily a solution.