Properties of polymers

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Sound propagation in solids

Longitudinal and transverse waves can propagate in solids. If the solid is rod-shaped or plate-shaped, bending waves that are not pure longitudinal or transverse waves are also possible.

Pressure waves are pure longitudinal waves. The deflection consists of a shortening or stretching of the rod. Torsional waves, in which the movement consists in twisting the rod sections against each other, are pure transversal waves.

Longitudinal and transverse waves propagate at different speeds. The torsional modulus (see shear modulus) is included in the propagation speed of transverse waves (torsional waves).

c_{trans} = \sqrt{\frac{G}{ρ}} \begin{array}{l} c_{trans} = Speed ​​of propagation of the transverse waves \\ G = Shear, shear, torsion modulus \\ ρ = Density \end{array}

Longitudinal waves, on the other hand, are associated with a one-sided compression of the rod. This is where the modulus of elasticity comes in.

c_{long} = \sqrt{\frac{E.}{ρ}} \begin{array}{l} c_{long} = Speed ​​of propagation of the longitudinal waves \\ E. = Modulus of elasticity \\ ρ = Density \end{array}

However, this equation is only valid as long as the transverse dimensions of the rod are small compared to the wavelength. The theory gives the following expression for the velocity of the longitudinal wave in an infinitely extended medium:

c_{long} = \sqrt{\frac{E.}{ρ} \cdot \frac{1 - μ}{(1 + μ) (1 - 2 μ)}} \begin{array}{l} c_{long} = Speed ​​of propagation of the longitudinal waves \\ E. = Modulus of elasticity \\ ρ = Density \\ μ = Poisson's number \end{array}

Tab. 1: Velocities of sound in different media

material	longitudinal in $m s^{-1}$	transversal in $m s^{-1}$
air	346	-
Ethanol	1207	-
water	1497	-
Polyethylene	920	540
Polystyrene	2240	1120
Lead, soft	1190	700
Copper, soft	3810	2325
Iron, 99.8%	5200	3240
beryllium	12870	8880

Air and liquids at 25 $° C$ . The speed of longitudinal waves in solids is related to the formula (finite expansion of the solid). The speeds of sound in solids are dependent on their pretreatment.

The speed of propagation of flexural waves is dependent on high frequency. The higher the frequency, the faster the waves propagate. Because of this dispersion, the sound signals change on the way from the source to the receiver. If, for example, the ice layer of a frozen body of water is hit briefly, an observer who is sufficiently distant will hear a "piuuh" instead of a hit, i.e. first high and then lower frequencies in quick succession.