Waves and Hearing

Wave speed on a stretched string

$v = \displaystyle\sqrt{\frac{F_T}{\mu}}$ ,

where

$F_T$ is the tension in the string, $[N]$;
$\mu$ is the mass per length of the string, $[kg/m]$.

Periodic waves speed

$v = \displaystyle\lambda⋅f$ ,

where

$\lambda$ is the wave length, $[m]$;
$f$ is the wave frequency, $[Hz]$.

Sinusoidal wave function

$y(x,t)=A⋅sin(k⋅x-\omega ⋅t - \phi)$ ,

where

$A$ is the value of wave amplitude, $[m]$;
$k=\displaystyle\frac{2\pi}{\lambda}$ is the wave number, $[rad/m]$;
$x$ is the position, $[m]$;
$ω=\displaystyle {2\pi}/{T}$ is the angular frequency, $[rad/s]$;
$T=\displaystyle 1/f$, is the wave period, $[s]$;
$t$ is the time, $[s]$;
$\phi$ is the wave phase, $[rad]$.

Transverse speed of a periodic wave

$v_y=-\omega ⋅A⋅cos(k⋅x-\omega⋅t)$ ,

where

$v_y$ is the trasversal wave speed, $[m/s]$;
$ω$ is the angular frequency, $[rad/s]$;
$A$ is the value of wave amplitude, $[m]$;
$k$ is the wave number, $[rad/m]$;
$t$ is the time, $[s]$.

Transverse acceleration of a periodic wave

$a_y=-\omega^2 ⋅A⋅sin(k⋅x-\omega⋅t)$ ,

where

$a_y$ is the trasversal wave acceleration, $[m/s^2]$;
$ω$ is the angular frequency, $[rad/s]$;
$A$ is the value of wave amplitude, $[m]$;
$k$ is the wave number, $[rad/m]$;
$t$ is the time, $[s]$.

Wave power on a stretched string

$W=\displaystyle \frac{1}{2}⋅\left[\mu⋅(\omega⋅A)^2\right]⋅v$ ,

where

$\mu$ is the linear dentity of the stretched string, $[kg/m]$;
$ω$ is the angular frequency, $[rad/s]$;
$A$ is the value of wave amplitude, $[m]$;
$v$ is the wave speed, $[m/s]$.

Beat (acoustics)

$f_{beat} = |f_1-f_2|$ ,

where

$f_{beat}$ is the beat frequency, $[Hz]$;
$f_1$ is the frequency of wave 1, $[Hz]$;
$f_2$ is the frequency of wave 2, $[Hz]$.

Doppler effect

$f_{observed} = \displaystyle f_0⋅\left(\frac{c±v_r}{c±v_s}\right)$ ,

where

$f_{observed}$ is the observed frequency, $[Hz]$;
$f_0$ emitted frequency, $[Hz]$;
$c$ is the propagation speed of waves in the medium, $[m/s]$;
$v_r$ is the speed of the receiver relative to the medium, added to $c$ if the receiver is moving towards the source, subtracted if the receiver is moving away from the source, $[m/s]$;
$v_s$ is the speed of the source relative to the medium, added to $c$ if the source is moving away from the receiver, subtracted if the source is moving towards the receiver, $[m/s]$.