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(kx  t), at the instant for time t=0. a b c d e i) Which particle moves according to y=A cos(  t) ? ii) Which particle moves according to y=A sin(  t) ? iii) If y e (t)=A cos(  t+  e ) for particle e, what is  e ? Assume y e (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.) (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 Velocity

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 Velocities Transverse particle displacement, y (x,t) Transverse particle velocity, ( x held constant) this is called the transverse velocity (Note that v y is not the wave speed v ! ) Transverse acceleration,

“Standard” Traveling sine wave (harmonic wave): maximum transverse displacement, y max = A maximum transverse velocity, v max =  A maximum transverse acceleration, a max =  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) v y (x, t) and maximum transverse speed a y (x, t) and maximum transverse acceleration v wave

Solution

Energy density in a wave dx ds dm 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 mv 2 ): The total energy per unit length is = 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 Both the energy density and the power transmitted are proportional to the square of the amplitude and square 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 ?