Lecture 25: Waves

Mechanical Waves;

travel within a medium/material, string in water or air

Transverse Waves;

string = perpendicular to wave, wave = x, amplitude = y

Longitudinal waves

fluid/medium = parallel to wave, medium = x, wave = x

Transverse and Longitudinal waves

ex. waves on surface of fluid wave = x , particles = circle

Periodic Waves

sinusoidal waves, each particle is in SHM

Wave Speed (v)

speed of propagation, how fast a given crest/peak in wave is travelling, v = wavelength f

1D Wave equation:

A²y / Ax² = 1/v² A²y / At²

!D wave equation solution:

y = A cos (kx +- wt + o) (- = +x)

wave number (k)

k = 2pi/wavelength

Angular frequency (w)

w = 2pi f = vk

phase velocity (v)

v = w/k → travelling velocity of a particular point on the wave

in order to travel with a fixed point on the wave, the value of cos

must be a _____

constant, kx +- wt + o = constant

Transverse Velocity (1st time derivative of wave function)

v = Ay/At = Aw sin (kx - wt + o)

Transverse Acceleration (2nd time derivative of wave function)

a = A²y/At² = -Aw² cos (kx - wt + o) = -w²y

Transverse velocity and acceleration are _____

partial derivatives

curvature can be found by taking _______s of

the ______

2 position derivatives, wave function

Curvature in String formula

A²y/At² = -Ak²cos(kx - wt+ + o) = -k²y

Wave Function definition:

describes wave trajectory

wavelength definition:

linear length of one complete cycle along acis

in wave function, if a wave is travelling to the right the sign will be __, why?

negative, cos = slightly shifted to the right (+ve x)

as t inc, wave moves in the ____ direction

negative

speed of wave (v) formula

v = wavelength f, sqrt(T /u)

frequency formula

f = 1/T

Energy transported by wave on a string function:

P = sqrt(uT) w²A²sin²(kx-wt)

Power function (Watts) definition:

energy transport per second at a given position and time on string

Maximum power function:

P max = sqrt(uT) w²A²

Average power function:

sin part = ½ , P avg =1/2 sqrt(uT) w²A²

For waves transported out in 3D average intensity I:

I = P/4pir²