EP5 Note No. 4 Wave Motion 4. Sinusoidal Waves, Wave Number Waves propagate in space in contrast to oscillations which are con ned in limited regions. In describing wave motion, spatial coordinates enter as independent variables in addition to time. To illustrate creation of wave motion in a string which is subject to tension, we give sinusoidal perturbation at one end. As time progresses, sinusoidal waves start propagating along the string as shown in the snapshots. The spatial period (m), called wavelength, depends on the frequenc of sinusoidal perturbation (Hz). The wavelength is inversel proportional to the frequenc and their product is constant and equal to the wave velocit c w (m s ), = c w () The wave velocit is a characteristic propert of the string as we will stud later. The rst series of snapshots is at a high frequenc and the second series is at a lower frequenc (one half). Note that the wavelength at the lower frequenc is doubled. A sinusoidal wave is periodic in both time t and spatial coordinate. For an observer who is using an oscilloscope to displa detected wave, a sinusoidal wave form appears on the screen as a function of time. Alternativel, the observer can use a fast camera to tae a snapshot of the spatial, sinusoidal wave pattern as a function of spatial coordinate. B successivel displaing man snapshots, a movie of moving wave motion can be constructed. sin ( ) 3 4 5 sin ( :) 3 4 5 sin ( :4) 3 4 5 sin ( :6) 3 4 5
sin ( :8) 3 4 5 Suppose at t = ; a snapshot of a sinusoidal wave is taen, A sin at t = () After t seconds, the entire waveform is shifted b a distance c w t to the right because of wave propagation, A sin ( c wt) at arbitrar time (3) (Note that parallel shift of a function f() b a ields f( a):) Since time dependence is periodic with the frequenc ; the coe cient of the time variable in the sin function must be equal to ; c w = or = c w : (4) We introduce an important quantit = (rad m ) (5) called the wavenumber. The three quantities, ; and c w are related through A general epression for a sinusoidal wave is therefore = c w (6) A sin( t) (7) Eample : The velocit of transverse waves in a string is given b s T c w = l where T is the tension (N) and l is the linear mass densit (g/m) of the string. (We will derive this in Note 5.) (a) If T = 4 N and l = 5 g/m, what is the wave velocit? (b) A harmonic wave is ecited b a vibrator at frequenc = 5 Hz. Determine the wavelength and wavenumber : Solution. (a) The wave velocit is c w = q T l = q 4 N :5 g/m = 8:3 m/sec.
(b) The wavelength is and the wavenumber is = c w = = 8:3 m/sec 5 /sec = :89 m rad = = 3:33 rad m :89 m Eample : A certain displacement wave is described b (; t) = : sin[(:5 t)] (m): Determine the wavenumber, wavelength, wave frequenc ; and wave velocit. What is the epression for the velocit wave? Solution Comparing with standard form for a sinusoidal wave, A sin( t) = A sin t ; we readil nd the wavenumber = rad/m, the wavelength = m and frequenc = Hz. The velocit wave (not to be confused with wave velocit) is v(; t) = @ @t = : ( ) cos[(:5 t)] (m s ) 4. Wave Di erential Equation Undamped oscillations described b (t) = A sin t satisf the di erential equation, What equation does the wave function, d dt + = (8) (; t) = A sin( t) (9) satisf? Since (; t) is periodic in as well as t, we epect that the equation should contain a second order spatial derivative as well as second order time derivative. Let us calculate the second order spatial derivative of the function (; t) = A sin( t); @ (; t) = A cos( @ t) @ @ (; t) = A sin( t) () Note that di erentiation is partial, because the function (; t) depends on as well as t: Similarl, @ @t (; t) = A sin( t) () Therefore, the second order derivatives satisf the following equation, @ (; t) = @ @ (; t) @t 3
Recalling = c w ; we rewrite this in the form @ @ (; t) = (; t) () @ c w @t This is the wave di erential equation which we will derive for various waves, both mechanical and electromagnetic. The wave equation also allows solutions in the form (; t) = ( + c w t) (3) which describes waves propagating in the negative direction. Note that if a function f() is shifted to the left b a; a resultant function is f( + a): 4.3 Waves do not have to be harmonic The equation derived in the preceding section, @ @ (; t) = (; t) (4) @ c w @t allows solutions of arbitrar shape as long as (; t) is in the translational form (; t) = ( c w t) (5) Although we have derived the wave equation from a nown form of sinusoidal shape, waves do not have to be sinusoidal or harmonic. Proof is straightforward, Similarl, Therefore, @ @ ( c wt) = @( c wt) d @ dx (X) = @ @ ( c wt) = d dx (X) @ @t ( d c wt) = c w dx (X) d dx (X); X = c wt @ @ (; t) = (; t) @ c w @t The functional shape of ( c w t) can thus be arbitrar. It ma be harmonic, pulse, step function, etc. 4.4 Superposition of Waves, Beats, Phase and Group Velocities Although waves do not have to be sinusoidal, considering a single harmonic wave su ces in wave analsis because a wave form of arbitrar shape can be constructed b superposing man harmonic waves of di erent frequencies and wavelengths. This powerful mathematical technique is nown as Fourier analsis (or Fourier decomposition) and provides a basic principle for image reconstruction in holograph, -ra and NMR tomograph. In this section, we consider superposition of two harmonic waves of slightl di erent frequencies and wavelengths, A sin( t) + A sin[( + ) ( + )t]: (6) 4
For an observer measuring the wave as a function of time onl, the observing position can be arbitraril chosen, sa, at = ; A [sin(t) + sin[( + )t] = A sin + t cos t (7) where the formula sin + sin = sin + cos is used. If is small, Eq. (7) describes a harmonic wave sin(t) con ned between slowl varing envelopes, A cos t The case = : is shown below. sin (( + :5)t) cos( :5t)..5..5. 4 6 8 4 6 8 4 6 8 3 The amplitude of the harmonic wave is strongl modulated and the modulation frequenc is jj: (In this case there are ten oscillations between two neighboring envelope peas or nodes.) This phenomenon is called beat. If the frequencies of two waves are identical, there are no beats. A piano tuner uses this phenomenon to adjust the tone of a e using a tuning for. We now go bac to the general epression, A sin( t) + A sin[( + ) ( + )t] = A sin + + t cos t (8) As in the case of time domain, this equation describes spatial as well as temporal beats. Rapidl rippling function sin( t) is con ned in the envelopes, A cos t (9) which propagates at a speed Envelope speed = () If = = = = c w ; both ripples and envelopes propagate together at the same speed c w : In the limit ; ; the envelope velocit de nes the group velocit, c g = d d 5 ()
The velocit c p = () is called the phase velocit. For nondispersive waves, the phase and group velocities are identical, Nondispersive waves: = d d = c w; or = c w (3) where c w is constant without an dependence on or : For dispersive waves, Dispersive waves: 6= d d and the wave velocit depends on or : Tpical eamples of nondispersive waves are electromagnetic waves in free space and low frequenc sound waves in air. Water waves are strongl dispersive. Electromagnetic waves con ned in wave guides and light waves in optical bers are also dispersive. Nondispersive waves propagate without changing their wave forms, while dispersive waves become spatiall spread (i.e., dispersed) as the propagate. See Animation No. 7 in the Web site. The wave equation we have derived, @ @ (; t) = (; t) @ c w @t pertains to nondispersive waves onl, and dispersive waves obe more complicated equations. For a given wave di erential equation, the relationship (dispersion relation) between and can be found. Three tpical dispersion relations are schematicall shown. = ; p ; p + (4) omega 5 4 3 3 4 5 The straight line = c w is for a nondispersive wave. The parabolic curve shows the dispersion relation of long wavelength water wave which is approimatel given b = p g. The phase and group velocities are r g = ; d d = r g which impl that long wavelength modes propagate faster than short wavelength modes. The group velocit is one half the phase velocit. The hperbolic curve is for electromagnetic waves in waveguides and plasmas (ionized, conducting gases), = p c + (c) 6
where c is the cuto frequenc and c = = p " = 3 8 m/sec is the speed of light in free space. Eample 3 Find the phase and group velocities of waves described b the following dispersion relation, = p c + (c) (5) Solution The phase velocit is p = c + (c) There is nothing wrong with this epression. However, in most eperiments, the frequenc is the quantit that is controlled. Therefore, it is more convenient to write the phase velocit in terms of the frequenc, not wavenumber : Solving Eq. (5) for ; we nd p = c : c Then, = c p c = p c (c =) > c Note that the phase velocit is alwas larger than c: Wave energ is carried at the group velocit, and as long as the group velocit remains smaller than c; there is no contradiction to the relativit theor which asserts that nothing can travel faster than c: The group velocit can be found b di erentiating the dispersion relation once, = c + (c) d = c d d = c d = = cp ( c =) < c which indeed remains smaller than c: Eample 4 Consider superposition of two harmonic waves, sin( t) + sin(: :t) Since = = = and = = ; the wave is dispersive with the envelope propagating slower than the ripples. Draw several snapshots to illustrate that wave propagation is not simple parallel shift of the original waveform. Solution Snapshots at t = ; ; 4; 6; 8 are shown. Ripples travel a distance of spatial units in time units but the envelope travels slower. Note deformation in the wave form which is conspicuous at the nodes where the amplitude is small. sin( ) + sin(: : ) 7
3 4 5 sin( ) + sin(: : ) 3 4 5 sin( 4) + sin(: : 4) 3 4 5 sin( 6) + sin(: : 6) 3 4 5 sin( 8) + sin(: : 8) 3 4 5 8