Born of the Wave Equation

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Department of Mathematics & Geosciences Universit of Trieste

1 Corso di Laurea in Fisica - UNITS ISTITUZIONI DI FISICA PER IL SISTEMA TERRA Born of the Wave Equation FABIO ROMANELLI Department of Mathematics & Geosciences Universit of Trieste romanel@units.it

2 What is a wave? - Small perturbations of a stable equilibrium point Linear restoring force Harmonic Oscillation Coupling of harmonic oscillators the disturbances can propagate, superpose and stand Normal modes of the sstem

3 Derivation of the wave equation A B Consider a small segment of string of length Δx and tension F The ends of the string make small angles θ1 and θ with the x-axis. The vertical displacement Δ is ver small compared to the length of the string

4 A B F sin Resolving forces verticall 1 From small angle approximation sinθ ~ tanθ F = = θ F sin θ F (sin θ sin θ1) F F(tan θ tan θ1) The tangent of angle A (B) = pendence of the curve in A (B) given b

5 A B x x F F We now appl N to segment = µ Δ = t x ma F = µ Δ A B x x F t x µ F t = x ( ) B x ( ) A [ ] Δx

6 µ F t = [( x) B ( x) ] A Δx The derivative of a function is defined as f x = lim Δx 0 ( x) [ f(x + Δx) f ] Δx If we associate f(x+δx) with ( x) B x and f(x) with ( ) A as Δx -> 0 µ F t This is the linear wave equation for waves on a string = x

7 Solution of the wave equation Consider a solution of the form (x,t) = A sin(kx-ωt) = ω t Asin(kx ωt) = k x Asin(kx ωt) If we substitute these into the linear wave equation µ F ( ω Asin(kx ωt)) = µ F ω = k k Asin(kx ωt) and, using ω /k = F/µ = v, i.e. v = ω/k v = F µ x = v 1 t General form of LWE

8 Speed of waves A general propert of waves is that the speed of a wave depends on the properties of the medium, but is independent of the motion of the source of the waves. Consider a wave moving along a rope experimentall we find (i) the greater the tension in a rope the faster the waves propagate (ii) waves propagate faster in a light rope than a heav rope ie v tension (F) and v 1/mass known as Mersenne s law

This frequenc is: a) Inverse proportional to the length of the string (this was actuall known to the ancients,

9 Mersenne s law L Harmonie Universelle (1637) This book contains (Marine) Mersenne's laws which describe the frequenc of oscillation of a stretched string. This frequenc is: a) Inverse proportional to the length of the string (this was actuall known to the ancients, and is usuall credited to Pthagoras himself). b) Proportional to the square root of the stretching force, and c) Inverse proportional to the square root of the mass per unit length.

10 The linear wave equation Earlier we introduced the concept of a wavefunction to represent waves travelling on a string. All wavefunctions (x,t) represent solutions of the LINEAR WAVE EQUATION The wave equation provides a complete description of the wave motion and from it we can derive the wave velocit The most general solution is, for 1D homogeneous medium, (x,t)=g(x+vt)+f(x-vt)

D Alembert s solution D'Alembert (1747) "Recherches sur la courbe que forme une corde tendue mise en vibration" (Researches on the curve that a tense cord forms [when] set into

D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie roale des sciences et belles lettres de Berlin, vol.

11 D Alembert s solution D'Alembert (1747) "Recherches sur la courbe que forme une corde tendue mise en vibration" (Researches on the curve that a tense cord forms [when] set into vibration), Histoire de l'académie roale des sciences et belles lettres de Berlin, vol. 3, pages D'Alembert (1750) "Addition au mémoire sur la courbe que forme une corde tenduë mise en vibration," Histoire de l'académie roale des sciences et belles lettres de Berlin, vol. 6, pages x = x = ξ ξ x + η η x = ξ + η ; xx = (x, t) (ξ, η) with ξ=x-vt, η=x+vt ( ) = ξξ + ξη + ηη, tt = v ( ξξ ξη + ηη ) x x ξη = ξ η = ξ η = 0 = h(ξ) + g(η) (x, t) = h(x vt) + g(x + vt) and if the initial conditions are (x,0)=f(x) and initial velocit=0 (x, t) = 1 f(x vt) + f(x + vt)

12 Harmonic Waves A harmonic wave is sinusoidal in shape, and has a displacement at time t=0 = Asin π λ x A is the amplitude of the wave and λ is the wavelength (the distance between two crests); if the wave is moving to the right with speed v, the wavefunction at some t is given b: = Asin π λ (x vt)

13 Time taken to travel one wavelength is the period T Velocit, wavelength and period are related b = Asin π x λ t T The wavefunction shows the periodic nature of : at an time t has the same value at x, x+λ, x+λ and at an x has the same value at times t, t+t, t+t

14 It is convenient to express the harmonic wavefunction b defining the wavenumber k, and the angular frequenc ω This assumes that the displacement is zero at x=0 and t=0. If this is not the case we can use a more general form where φ is the phase constant and is determined from initial conditions

15 The wavefunction can be used to describe the motion of an point P. P Transverse velocit v which has a maximum value, (v ) max = ωa, when = 0

16 Transverse acceleration a P which has a maximum absolute value, (a ) max = ω A, when t=0 NB: x-coordinates of P are constant

17 Example A harmonic wave on a rope is given b the expression where the amplitude is in mm, k in rad m -1, and ω in rad s -1 (a) Determine the velocit and acceleration for each element of the rope. (b) What are the maximum values of the acceleration and velocit? (c) Is the displacement +ve or -ve at x=1m and t=0.s?

18 (a) Determine the velocit and acceleration for each element of the rope.

19 (b) What are the maximum values of the acceleration and velocit? (c) Is the displacement +ve or -ve at x=1m and t=0.s? Displacement is +ve

20 Energ of waves on a string Consider a harmonic wave travelling on a string. Δx, Δm Source of energ is an external agent on the left of the wave which does work in producing oscillations. Consider a small segment, length Δx and mass Δm. The segment moves verticall with SHM, frequenc ω and amplitude A. Generall E = 1 mω A

21 If we appl this to our small segment, the total energ of the element is If µ is the mass per unit length, then the element Δx has mass Δm = µ Δx If the wave is travelling from left to right, the energ ΔE arises from the work done on element Δm i b the element Δm i-1 (to the left).

22 Similarl Δm i does work on element Δm i+1 (to the right) ie. energ is transmitted to the right. The rate at which energ is transmitted along the string is the power and is given b de/dt. If Δx -> 0 then but dx/dt = speed

23 Power transmitted on a harmonic wave is proportional to (a) the wave speed v (b) the square of the angular frequenc ω (c) the square of the amplitude A All harmonic waves have the following general properties: The power transmitted b an harmonic wave is proportional to the square of the frequenc and to the square of the amplitude.

oscillators Linear restoring force the disturbances can propagate, superpose

24 What is a wave? - 3 Small perturbations of a stable equilibrium point Coupling of harmonic oscillators Linear restoring force the disturbances can propagate, superpose and stand Harmonic Oscillation WAVE: organized propagating imbalance, satisfing differential equations of motion 1 x = v t General form of LWE

Born of the Sound Wave Equation

Corso di Laurea in Fisica - UNITS ISTITUZIONI DI FISICA PER IL SISTEMA TERRA Born of the Sound Wave Equation FABIO ROMANELLI Department of Mathematics & Geosciences University of Trieste romanel@units.it