# What is the physical difference between compression waves and longitud

1. Dec 12, 2013

### mikelotinga

Hello, this is my first post on this forum so nice to be here, and I'll be very appreciative of any responses. My background is in acoustics, and hence my question is relevant to vibration propagation.

The terms 'compression' and 'longitudinal' are both frequently used to describe the same type of wave; namely one in which the direction of particle velocity is parallel to that of the group velocity.

However, in some texts I have seen a distinction drawn between the two, and two different formulae used to calculate the wave phase speed.

I will reproduce the two definitions in terms of the classical elastic constants sometimes known as the Lamé parameters $\lambda$ and $\mu$. I define these constants using Poisson's ratio $\nu$, and the elastic (Young's, wrongly attributed) modulus, $E$.

$$\lambda = \frac{\nu E}{(1 + \nu)(1 - 2\nu)}$$

$$\mu = \frac{E}{2(1 + \nu)}$$

For COMPRESSION (dilatational) waves, Ewing & Jardetsky, 1957 (Elastic Waves in Layered Media, available free at https://archive.org/details/elasticwavesinla032682mbp - see Chapter 1, pages 8 and 9 in particular) give the formula for the phase speed $c_P$ in terms of $\lambda$, $\mu$ and the material mass density $\rho$ as

$$c_P = \sqrt{\frac{\lambda + 2\mu}{\rho}}$$

For LONGITUDINAL waves, Cremer, Heckl and Petersson, 2005 (Structure-borne Sound, see this page and this page, noting that, in this volume, Poisson's ratio is denoted as $\mu$) give the formula for the phase speed $c_L$ as

$$c_L = \sqrt{\frac{E(1 - \nu)}{(1 + \nu)(1 - 2\nu)\rho}}$$

which is, in terms of $\lambda$

$$c_L = \sqrt{\frac{\lambda(1 - \nu)}{\nu\rho}}$$

The distinction between the compression wave phase speed and the longitudinal wave phase speed is therefore, as pointed out by Remington, Kurzweil and Towers, 1987 (Low-Frequency Noise and Vibration from Trains, Chapter 16 of the Transportation Noise Reference Book, edited by Paul Nelson, available here, see page 16/8, i.e. page 8 of chapter 16)

$$\frac{c_P}{c_L} = \sqrt{\frac{1 - \nu}{(1 + \nu)(1 - 2\nu)}}$$

Please note the longitudinal wave here is NOT the quasi-longitudinal wave found in relatively thin objects such as plates and bars, which have a different set of phase speeds again.

Now, 'compression' and 'longitudinal' appear to be bandied about as interchangeable terms for the same wave, but this analysis suggests this is not the case. Can anyone explain what the difference is in a physical, rather than mathematical sense?

Many thanks,

Mike

2. Dec 12, 2013

### tiny-tim

hello mike! welcome to pf!
so $$\lambda + 2\mu = \frac{E}{(1 + \nu)(1 - 2\nu)}(\nu + 1 - 2\nu)$$

3. Dec 12, 2013

### tiny-tim

hi mike!
what diffference??

i'm saying they're the same …​

4. Dec 12, 2013

### mikelotinga

Hi Tim,

Thanks for the welcome and for pointing out that, by the definitions of $c_P$ and $c_L$ in my post, in fact,

$$c_P = c_L$$

so there IS no mathematical distinction. I can see what happened. In the Remington text I mentioned, they use the definition

$$c_L = \sqrt{\frac{\mathbf{E}}{\rho}}$$

I've put the $E$ in bold because, since they neglect to define $\mathbf{E}$, I assumed (wrongly), that their $\mathbf{E}$ was the same as the $D$ used in the Cremer text, where $D = \lambda(1 - \nu)/\nu = \lambda + 2\mu$ as you've helpfully shown. Rather, working through from their ratio of $c_P/c_L$, it seems they DO actually mean $c_L = \sqrt{E/\rho}$ where $E$ IS the elastic modulus.

SO (deep breath), to rephrase the question, can anyone explain WHAT a so-called 'longitudinal' wave with a phase speed of

$$c = \sqrt{\frac{E}{\rho}}$$

is, and how it differs (physically) from the normal definition of a compression or longitudinal wave?

Thanks alot!

:)

5. Dec 12, 2013

### mikelotinga

Ah, got it, contrary to the bottom part of my original post, this last one IS the equation for the phase speed of a quasi-longitudinal wave in a bar.

Sorry to waste your time, puzzle solved!