PHY 102: Waves & Quanta Topic 2 Travelling Waves John 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 Combined)

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

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

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

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

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):

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

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……

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

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

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) T2T2 T1T1 y x T T T 2y T 1y x+∆x Small element of string ∆x motion

Wave Speed on a string Net force in y-direction: T 2y, T 1y given by: From Newton 2, :

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