You want the actual math or just quotes from the internet?

The general wave equation is described by a second order differential equation

y'' = 1/c^2 dy^2/dt^2

The prime denotes differentiation in respect with space. y is a state variable and usually denotes displacement. The constant c is the propagation speed of the wave in a medium.

The propagation speed depends on the material properties of the medium, and appears naturally in the wave equation when you calculate dynamic equilibrium for a system (force or energy equations).

To solve the wave equation you assume an harmonic solution of a propagating perturbarion. The most general solution is given by the D'Alambert solution that only states that a function of a coupled variable omega*t-kx is a solution. Omega is the angular frequency given by 2pi*freq and k is the wavenumber, which is the space frequency, inverse to the wavelength. This two quantities relates to the propagation velocity through

c = omega/k

Since omega = 2pi*freq and k=2pi/lambda , in which lambda = 2pi/wavelength, then c = lambda*frequency.

To obtain the wave speed in terms of the material properties for a specific wave, you have to look into its wave equation to see who forms the constant c.

For acoutic waves it depends if the medium is air or liquid. If its air, the wave speed is proportional to the medium pressure and mass density. If it is a incompressible liquid, it is given by the bulk modulus divided by the mass density. In general, the wave speed is proportional to a stiffness measure of the medium divided by its mass.

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