14Class 11 Physics

Waves formulas

Wave speed, strings, pipes and beats formulas. Draw the standing wave first, and the formula usually follows.

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All Physics formulas

Waves

Wave equation: $\displaystyle y = a\sin(kx - \omega t + \phi)$
Wave number and speed: $\displaystyle k = \frac{2\pi}{\lambda}, \quad v = \nu\lambda = \frac{\omega}{k}$
Speed on a string: $\displaystyle v = \sqrt{\frac{T}{\mu}}$
μ is mass per unit length.
Speed of sound (Laplace): $\displaystyle v = \sqrt{\frac{\gamma P}{\rho}}$

Standing waves

String fixed at both ends: $\displaystyle \nu_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \dots$
Open pipe: $\displaystyle \nu_n = \frac{nv}{2L}, \quad n = 1, 2, 3, \dots$
Closed pipe: $\displaystyle \nu_n = \frac{(2n - 1)v}{4L}, \quad n = 1, 2, 3, \dots$
Only odd harmonics exist.

Beats

Beat frequency: $\displaystyle \nu_{\text{beat}} = |\nu_1 - \nu_2|$

Where students slip up

Using even harmonics for a closed pipe, which only has odd ones.
Using the tension in the formula without converting μ. It's mass per unit length, not total mass.
Thinking frequency changes when a wave enters a new medium. Frequency stays the same, while speed and wavelength change.

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