Wavelength of a mechanical wave in different media

Question

I quote the following sentence from a book:

"Two mechanical waves with the same amplitude, wavelength, and frequency will move at different speeds if they are moving through different materials."

The confusion is whether to depend on the equation $v=\lambda*f$; this implies that waves will have the same speed if they have the same frequency and wavelength.

The other misunderstanding is when using $ v = \sqrt{(F/\mu)}$; which implies that the speed of a wave is independent of its wavelength and frequency but sololy depends on the linear density and elasticity of the material.

So the question is how does the speed of a mechanical wave change in different media and on what factors does it depend? Is the quote correct?

Answer 1

The quote seems to me to not be entirely correct as the wavelength of two waves of the same frequency is not the same in different mediums. As you correctly stated

$v=\sqrt{\dfrac{F}{\mu}}$

This can be derived from the wave equation which examines the curvature and transverse acceleration of the wave. As a result of this, the velocity must change when the wave propagates into a new medium. Now if you link this back to the equation

$v=\lambda f$

You could get the impression that either the wavelength or the frequency of the transmitted wave could be different from the parent wave to accommodate the velocity change. However, there are restrictions on the behavior of the waves at the boundary between the media. Their transverse positions as well as their curvatures must be equal at the point of transition at any point in time. This means that the frequencies can not be different from each other because the boundary conditions would not be satisfied all the time. Hence it must be the wavelength that changes at a boundary.

In the general case where the waves in two different media are not related to one another, it still holds true that either the frequency or the wavelength must be different to allow for a different velocity but there are no longer boundary conditions restricting the frequency change.

Answer 2

The velocity v in wavelength equation can be calculated as a square root of elastic modulus devided by material density. Velocity obtained from this equation is called transversal speed of sound and it governs resonance of the vibrating body.