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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. y. A. x. -A. Remember: the particles in the medium move vertically.

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Wave Motion II

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  1. Wave Motion II • Sinusoidal (harmonic) waves • Energy and power in sinusoidal waves

  2. For a wave traveling in the +x direction, the displacement y is given by y (x,t) = A sin (kx – t) with  = kv y A x -A Remember: the particles in the medium move vertically.

  3. 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: y = A sin (kx – wt) = A sin [ constant – wt] The “angular frequency” of the particle motion is w; the initial phase is kx (different for different x, that is, particles). ω = 2πf ω=“angular frequency” radians/sec f =“frequency” cycles/sec (Hz=hertz)

  4. y a A Example e b x d c -A Shown is a picture of a travelling wave, y=A sin(kx- wt), at the instant for time t=0. i) Which particle moves according to y=A cos(wt) ? ii) Which particle moves according to y=A sin(wt) ? iii) If ye(t)=A cos(wt+fe ) for particle e, what is fe ? Assume ye(0)=1/2A

  5. Wave Velocity The wave velocity is determined by the properties of the medium; for example, 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)

  6. Example 1: What are w, k and for a 99.7 MHz FM radio wave?

  7. 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.

  8. Transverse 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,

  9. “Standard” Traveling sine wave (harmonic wave): maximum transverse displacement, ymax = A maximum transverse velocity, vmax = w A maximum transverse acceleration, amax = w 2 A These are the usual results for S.H.M

  10. Example 3: y string: 1 gram/m; 2.5 N tension x vwave Oscillator: 50 Hz, amplitude 5 mm, y(0,0)=0 Find: y (x, t) vy (x, t)and maximum transverse speed ay (x, t) and maximum transverse acceleration

  11. Solution

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

  13. 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 = energy density Where does the potential energy in the string come from?

  14. Power transmitted by harmonic wave with wave speed v: A distance v of the wave travels past a fixed point in the string in one second. 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.

  15. 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 ?

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