Average power of sinusoidal wave on stretched string

The average power of a wave of any type is proportional to the square of its amplitude and to the square of its angular frequency.

Conditions:

1. The wave is sinusoidal.

2. The wave is linear, i.e., assuming \(s(x, t)\) is the transverse displacement:

   1. \(\frac{|s(x, t)|}{L} \ll 1\), where \(L\) is the string length

   2. \(\left| \frac{\partial s}{\partial x} \right| \ll 1\)

   3. the tension is constant thoughout the string

   4. the linear density is constant throughout the string

   5. the displacement must be orthogonal to the string’s equilibrium shape

   6. no damping or energy dissipation within the string

Links:

1. Physics LibreTexts, formula 16.5.1.

wave_average_power

Average power, or rate of energy transfer, of the wave.

Symbol:

P

Latex:

\(P\)

Dimension:

power

string_linear_density

linear_density of the string.

Symbol:

mu

Latex:

\(\mu\)

Dimension:

mass/length

phase_speed

phase_speed of the wave.

Symbol:

v

Latex:

\(v\)

Dimension:

velocity

wave_angular_frequency

angular_frequency of the wave.

Symbol:

w

Latex:

\(\omega\)

Dimension:

angle/time

wave_amplitude

Amplitude of the wave. See euclidean_distance.

Symbol:

u_max

Latex:

\(u_\text{max}\)

Dimension:

length

law

P = mu * v * w^2 * u_max^2 / 2

Latex:

\[P = \frac{\mu v \omega^{2} u_\text{max}^{2}}{2}\]