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Wave Motion

Wave Motion. Qualitative properties of wave motion Mathematical description of waves in 1-D Sinusoidal waves. A wave is a moving pattern . For example, a wave on a stretched string:. v. time t. Δ x = v Δ t. t +Δ t.

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Wave Motion

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  1. Wave Motion • Qualitative properties of wave motion • Mathematical description of waves in 1-D • Sinusoidal waves Physics 1B03summer-Lecture 9

  2. A wave is a moving pattern. For example, a wave on a stretched string: v time t Δx = v Δt t +Δt The wave speed v is the speed of the pattern. No particles move at this wave speed – it is the speed of the wave. However, the wave does carry energy and momentum. Physics 1B03summer-Lecture 9

  3. Transverse waves The particles move up and down The wave moves this way If the particle motion is perpendicular to the direction the wave travels, the wave is called a “transverse wave”. Examples: Waves on a string; waves on water; light & other electromagnetic waves; some sound waves in solids (shear waves) Physics 1B03summer-Lecture 9

  4. Longitudinal waves The wave moves long distances parallel to the particle motions. The particles back and forth. Example: sound waves in fluids (air, water) Even in longitudinal waves, the particle velocitiesare quite different from the wave velocity. The speed of the wave can be orders of magnitude larger than the particle speeds. Physics 1B03summer-Lecture 9

  5. Quiz B A C wave motion Which particle is moving at the highest speed? Physics 1B03summer-Lecture 9

  6. Non-dispersive waves: the wave always keeps the same shape as it moves. For these waves, the wave speed is determined entirely by the medium, and is the same for all sizes & shapes of waves. eg. stretched string: (A familar example of a dispersive wave is an ordinary water wave in deep water. We will discuss only non-dispersive waves.) Physics 1B03summer-Lecture 9

  7. Principle of Superposition When two waves meet, the displacements add: (for waves in a “linear medium”) So, waves can pass through each other: v v Physics 1B03summer-Lecture 9

  8. Quiz: “equal and opposite” waves v v v v Sketch the particle velocities at the instant the string is completely straight. Physics 1B03summer-Lecture 9

  9. Reflections Waves (partially) reflect from any boundary in the medium: 1) “Soft” boundary: light string, or free end Reflection is upright Physics 1B03summer-Lecture 9

  10. Reflections 2) “Hard” boundary: heavy rope, or fixed end. Reflection is inverted Physics 1B03summer-Lecture 9

  11. The math: Suppose the shape of the wave at t = 0 is given by some function y = f(x). y = f (x) at time t = 0: y v vt at time t: y = f (x - vt) y Note: y = y(x,t), a function of two variables; f is a function of one variable Physics 1B03summer-Lecture 9

  12. Non-dispersive waves: y (x,t) = f (x ± vt) + sign: wave travels towards –x - sign: wave travels towards +x f is any (smooth) function of one variable. eg. f(x) = A sin (kx) Physics 1B03summer-Lecture 9

  13. 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

  14. y Quiz v x A wave y=f(x-vt) travels along the x axis. It reflects from a fixed end at the origin. The reflected wave is described by: • y= - f(x-vt) • y= - f(x+vt) • y= - f(- x-vt) • y= - f(- x+vt) • something else Physics 1B03summer-Lecture 9

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