The velocity of any mechanical wave, whether transverse or longitudinal, depends both on the inertial properties of the medium (to store kinetic energy) and its elastic properties (to store potential energy). Thus, we can write, generically, that the velocity of a wave is given by:

\(v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}} \)

The speed of a wave is independent of the movement of the source relative to the medium.

Speed on a Rope

It is given by

\( v = \sqrt{ \frac{T}{\mu} } \)

Where:
\(v\) = wave velocity in the rope
\(T\) = tension in the rope
\(\mu\) = linear density of the string

Sound Speed in an Ideal Gas

\( v = \sqrt{ \frac{RT}{M} } \)
\(R\) = Gas constant
\(T\) = Absolute temperature
\(M\) = molecular weight

Wave Function

A wave function y(x, t) describes the displacement of individual particles from the medium. For a sine wave traveling in the positive x direction, we have:

\( y(x , t) = A sin (kx - \omega t) \)
\( y(x , t) = A sin ( \omega t - \frac{x}{v}) \)
\( y(x , t) = A sin ( 2 \pi ft - \frac{x}{v}) \)
\( y(x , t) = A sin ( 2 \pi \frac{t}{T} - \frac{x}{\lambda}) \)
Where:
\( A = y_{max} \) , the wave amplitude
\( k = \frac{2 \pi}{\lambda} \), the wave number
\( \omega \) = the angular frequency

Transmitted Power

The power transmitted by any harmonic wave is proportional to the square of the frequency and to the square of the amplitude. On a string, the expression is:

\( P = \frac{\mu \omega^2 A^2 v}{2} \)

Relations

  • \( f = \frac{1}{\tau} \)
  • \( v = \lambda f \)
  • \( v = \frac{\lambda}{\tau} \)
  • \( \omega = \frac{2 \pi}{\tau} \)
  • \( \omega = 2 \pi f \)