mikelotinga
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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
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
