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Wave Motion

Wave Motion. Sinusoidal waves Energy and power in sinusoidal waves. Sine Waves. For sinusoidal waves, the shape is a sine function, eg .,. f(x) = y(x,0) = A sin (kx). ( A and k are constants). y. A. x. -A. Then y (x,t) = f(x – vt) = A sin[ k(x – vt) ].

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Wave Motion

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  1. Wave Motion • Sinusoidal waves • Energy and power in sinusoidal waves Physics 1B03summer-Lecture 9

  2. Sine Waves For sinusoidal waves, the shape is a sine function, eg., f(x) = y(x,0) = A sin(kx) (A and k are constants) y A x -A Then y (x,t) = f(x – vt) = A sin[k(x – vt)] Physics 1B03summer-Lecture 9

  3. y (x,t) = A sin[kx – wt] Sine wave: y l A v x -A l(“lambda”)is the wavelength (length of one complete wave); and so (kx) must increase by 2π radians (one complete cycle) when x increases by l. So kl = 2p, or k = 2π / λ Physics 1B03summer-Lecture 9

  4. Rewrite:y = A sin [kx – kvt]=A sin [kx – wt] The displacement of a particle at location x is a sinusoidal function of time – i.e., simple harmonic motion: y = A sin [ constant – wt] The “angular frequency” of the particle motion is w=kv; the initial phase is kx (different for different particles). Review: SHM is described by functions of the form y(t) = A cos(wt+f) = A sin(p/2 –f –wt), etc., with ω = 2πf “angular frequency” radians/sec frequency: cycles/sec (=hertz) Physics 1B03summer-Lecture 9

  5. y a A Quiz e b x d c -A Shown is a picture of a wave, y=A sin(kx- wt), at time t=0 . i) Which particle moves according to y=A cos(wt) ? ii) Which particle moves according to y=A sin(wt) ? iii) Sketch a graph of y(t) for particle e. Physics 1B03summer-Lecture 9

  6. The most general form of sine wave is y = Asin(kx ± ωt – f) amplitude “phase” y(x,t) = A sin (kx ± wt –f ) phase constant f angular wavenumber k = 2π/ λ (radians/metre) angular frequency ω = 2πf (radians/second) The wave speed is v = 1 wavelength / 1 period, so v = fλ = ω / k Physics 1B03summer-Lecture 9

  7. Thus, y = A sin (kx ± ωt) or in general y = Asin(kx ± ωt-) A: amplitude k = 2π/λ : “angular wavenumber” (radians/metre) ω = 2πf : “angular frequency” (radians/second) : “phase constant” (radians) Note: Wave speed, v = 1 wavelength / 1 period v = f λ = ω/k Physics 1B03summer-Lecture 9

  8. Wave Velocity The wave velocity is determined by the properties of the medium: Transverse waves on a string: (proof from Newton’s second law) Electromagnetic wave (light, radio, etc.): v = c 2.998108 m/s(in vacuum) v = c/n(in a material with refractive index n) (proof from Maxwell’s Equations for E-M fields) Physics 1B03summer-Lecture 9

  9. Exercise What are w and k for a 99.7 MHz FM radio wave? Physics 1B03summer-Lecture 9

  10. Particle Velocities • Particle displacement, y (x,t) • Particle velocity, vy = dy/dt (x held constant) • (Note that vy is not the wave speed v – different directions! ) • Acceleration, Physics 1B03summer-Lecture 9

  11. “Standard” sine wave: maximum displacement, ymax = A maximum velocity, vmax = w A maximum acceleration, amax = w 2 A Same as before for SHM ! Physics 1B03summer-Lecture 9

  12. Example y string: 1 gram/m; 2.5 N tension x vwave Oscillator: 50 Hz, amplitude 5 mm Find: y (x, t) vy (x, t)and maximum speed ay (x, t) and maximum acceleration Physics 1B03summer-Lecture 9

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