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PHY 102: Waves & Quanta Topic 2 Travelling Waves John Cockburn (j.cockburn@... Room E15)

PHY 102: Waves & Quanta Topic 2 Travelling Waves John Cockburn (j.cockburn@... Room E15). What is a wave? Mathematical description of travelling pulses & waves The wave equation Speed of transverse waves on a string. TRANSVERSE WAVE. LONGITUDINAL WAVE. WATER WAVE (Long + Trans

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PHY 102: Waves & Quanta Topic 2 Travelling Waves John Cockburn (j.cockburn@... Room E15)

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  1. PHY 102: Waves & Quanta Topic 2 Travelling Waves John Cockburn (j.cockburn@... Room E15)

  2. What is a wave? • Mathematical description of travelling pulses & waves • The wave equation • Speed of transverse waves on a string

  3. TRANSVERSE WAVE LONGITUDINAL WAVE WATER WAVE (Long + Trans Combined)

  4. Disturbance moves (propagates) with velocity v (wave speed) • The wave speed is not the same as the speed with which the particles in the medium move • TRANSVERSE WAVE: particle motion perpendicular to direction of wave propagation • LONGITUDINAL WAVE: particle motion parallel/antiparallel to direction of propagation • No net motion of particles of medium from one region to another: WAVES TRANSPORT ENERGY NOT MATTER

  5. f(x) f(x+5) f(x-10) Mathematical description of a wave pulse GCSE(?) maths: Translation of f(x) by a distance d to the rightf(x-d) For wave pulse travelling to the right with velocity v : f(x) f(x-vt) d=vt function shown is actually: 0

  6. Sinusoidal waves Periodic sinoisoidal wave produced by excitation oscillating with SHM (transverse or longitudinal) Wavelength λ Every particle in the medium oscillates with SHM with the same frequency and amplitude

  7. Sinusoidal travelling waves: particle motion Disturbance travels with velocity v Travels distance λ in one time period T

  8. Sinusoidal travelling waves: Mathematical description Imagine taking “snapshot” of wave at some time t (say t=0) Dispacement of wave given by; If we “turn on” wave motion to the right with velocity v we have (see slide 5):

  9. Sinusoidal travelling waves: Mathematical description We can define a new quantity called the “wave number”, k = 2/λ NB in wave motion, y is a function of both x and t

  10. The Wave Equation Curvature of string is a maximum Particle acceleration (SHM) is a maximum Curvature of string is zero Particle acceleration (SHM) is zero So, lets make a guess that string curvature  particle acceleration at that point……

  11. The Wave Equation Mathematically, the string curvature is: And the particle acceleration is: So we’re suggesting that:

  12. The Wave Equation Applies to ALL wave motion (not just sinusoidal waves on strings)

  13. T2y T2 y motion T Small element of string T T1 T1y ∆x x+∆x x Wave Speed on a string Small element of string (undisturbed length ∆x) undergoes transverse motion, driven by difference in the y-components of tension at each end (x-components equal and opposite)

  14. Wave Speed on a string Net force in y-direction: T2y, T1y given by: From Newton 2, :

  15. Wave Speed on a string Now in the limit as ∆x0: So Finally: Comparing with wave equation:

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