Wave Motion II Wave Motion II

•Sinusoidal (harmonic) waves

•Energy and power in sinusoidal waves

For a wave traveling in the +x direction, the displacement y is given by

y (x,t) = A sin (kx – t) with = kv

A

-A

y

x

Remember: the particles in the medium move vertically.

y = A sin (kx – t) = A sin [ constant – t]

ω = 2πf

ω=“angular frequency”radians/sec

f =“frequency” cycles/sec (Hz=hertz)

The transverse displacement of a particle at a fixed location x in the medium is a sinusoidal function of time – i.e., simple harmonic motion:

The “angular frequency” of the particle motion is ; the initial phase is kx (different for different x, that is, particles).

Example

A

-A

y

x

Shown is a picture of a travelling wave, y=A sin(kxt), at the instant for time t=0.

a

b

cd

e

i) Which particle moves according to y=A cos(t) ?

ii) Which particle moves according to y=A sin(t) ?

iii) If ye(t)=A cos(t+e ) for particle e, what is e ? Assume ye(0)=1/2A

1) Transverse waves on a string:

(proof from Newton’s second law and wave equation, S16.5)

2) Electromagnetic wave (light, radio, etc.)

wave

tension Tmass/unit length

v

cv oo

1

(proof from Maxwell’s Equations for E-M fields, S34.3)

The wave velocity is determined by the properties of the medium; for example,

Wave Wave VelocityVelocity

Example 1:

What are , k and for a 99.7 MHz FM radio wave?

Example 2:

A string is driven at one end at a frequency of 5Hz. The amplitude of the motion is 12cm, and the wave speed is 20 m/s. Determine the angular frequency for the wave and write an expression for the wave equation.

Transverse Particle VelocitiesTransverse Particle Velocities

Transverse particle displacement, y (x,t)

Transverse particle velocity, (x held

constant) this is called the transverse velocity

(Note that vy is not the wave speed v ! )

Transverse acceleration, 2

2

y

y

v ya

t t

y

yv

t

2

2

sin( )

cos( )

sin( )

y

y

y

y A kx ty

v A kx ttv

a A kx tt

y

“Standard” Traveling sine wave (harmonic wave):

maximum transverse displacement, ymax = A maximum transverse velocity, vmax = A maximum transverse acceleration, amax = 2 A

These are the usual results for S.H.M

Example 3:

string: 1 gram/m; 2.5 N tension

Oscillator:50 Hz, amplitude 5 mm, y(0,0)=0

y

x

Find: y (x, t)vy (x, t) and maximum transverse speed

ay (x, t) and maximum transverse acceleration

vwave

Solution

Energy density in a wave

dx

dsdm

Ignore difference between “ds”, “dx” (this is a good approx for small A, or large ): dm = μ dx (μ = mass/unit length)

Each particle of mass dm in the string is executing SHM so its total energy (kinetic + potential) is (since E= ½ mv2):

The total energy per unit length is

2 2 2 21 1 dx

2 2dE dm A A

2 212

dEA

dx = energy density

Where does the potential energy in the string come from?

Power transmitted by harmonic wave with wave speed v:

So:

For waves on a string, power transmitted is

P energy density v

vAP 2221

Both the energy density and the power transmitted are proportional to the square of the amplitude andsquare of the frequency. This is a general property of sinusoidal waves.

A distance v of the wave travels past a fixed point in the string in one second.

Example 4:

A stretched rope having mass per unit length of μ=5x10-2 kg/m is under a tension of 80 N. How much power must be supplied to the rope to generate harmonic waves at a frequency of 60 Hz and an amplitude of 6cm ?