4 Spring 99 Problem Set Optional Problems Physics February, 999 Handout Sinusoidal Waes. sinusoidal waes traeling on a string are described by wae Two Waelength is waelength of wae?ofwae? In terms of amplitude A, period T and waelength, of a sinusoidal traeling wae isy(x; t) Acos, ( x, t T ). form 5 Frequency is frequency of wae?ofwae? Using same expression as in part (, we hae for wae f 3 T f T relation between frequency f, waelength, and wae for a sinusoidal wae isf. Sowehae speed f 4:77 :6 cm :6 cm f :9 :34 cm :6 cm both waes are traeling on same string, y must hae Since speed. same Phase Dierence d) is phase dierence between waes? phase of wae is 5x,3t; phase of wae x,t. phase dierence is is 5x, 3t, (x, t) 3x,8t Superposition e) superposition of se waes is y3(x; t) y(x; t)+y(x; t). Find an relating alues of x and t for which superposition y3(x; t) equation always zero. (This is called destructie interference). is y(x; t) cos(5x, 3t) and y(x; t) cos(x, t). In functions equations, y and x are in cm and t is in. se Speed c) is speed of waes on string? Comparing this expression with that of wae,we see that or x 5x cm:6 cm For wae,we hae x x cm:34 cm, t 3t T Hz 4:77 Hz For wae,we hae t t T Hz :9 Hz
a phase shift of n into a wae, where n is an odd integer, Introducing sign of wae. Since both wae and wae hae same changes sum of two waes will be zero wheneer phase amplitude, is relation between x and t which makes sum of two waes This zero. Speed of Transerse Waes. aluminum wire has 3 tions as shown. Section has radius r, tion An has radius r, and tion 3 has radius 3r. are ratios of of a transerse wae propagating in three tions? speeds : : 3 3 : : (A) 9 : 4 : (B) : / : /3 (C) elocity of wae is related to mass per unit, and tension,, by: p.for a wire of density and length, r, r. We see that / r. refore, : : 3 r : radius :3r::3 and correct answer is (C). r Pulse on a String 3. plot shows a pulse on a string at time t. wae speed is quantitatiely transerse elocity (\chunk speed") of Sketch as a function of x at t. Label your axes clearly and be sure to string use relation between transerse elocity and slope appropriate We dierence between waes is n. Thus we hae 3x,8t n ms,. p : p 3 (D): : : 3 (E) gie units. for a traeling wae moing in +x direction: y y, From this relation and graph gien aboe we hae:
If wae speed is 5m/s, describe in detail motion which should gien to end of string to produce this pulse. be steep part of pulse must hae been made rst since is farst from source. waeform traels.5 m horizontally it 5m/s in going from minimum to maximum displacement so this must at / of a ond. wae n traels.5 m horizontally at 5m/s take going from maximum to minimum displacement which takes / of a in refore, to produce waeform shown, end of string ond. be raised cm at a uniform elocity of cm/.s m/s and must lowered cm at a uniform elocity of cm/.5s.m/s. n How is answer changed if wae speed is m/s? If wae speed is now m/s, n to produce waeform indicated, end of string must be raised cm at a uniform of cm/.5s m/s and n lowered cm at a uniform elocity elocity of cm/.5s.4m/s. Wae Propagation 5. time t awae pulse looks as shown. It is propagating in +x At Propagation in t does pulse look like as a function of x at time.5 onds later? a At a time.5s later, pulse will hae traeled d t 5 along +x axis, so it will look same but be translated 5 m. This m Wae Pulses 4. want to produce a wae pulse on a string that is moing to right You direction with elocity m/s..8 and that has form shown in gure..6.4 - -5 5 x (m) is shown below: 3
Propagation in x does pulse look like as a function of time x m? at This is trickier. One good way to handle this is to make marks on original cure showing time at which that part of tick wae will cross x. For example, on original axis at x you put a tick mark labeled s, at x, myou would put a tick would labeled s, and so on, because those are times when that point mark cure gets to x. You can n redraw graph with y s. t on Note that pulse looks reersed since leading edge will reach axes. point onxaxis at earlier times relatie to trailing edge. a Wae Equation 6. a wae described by y(x; t) f(x, t) that is traelling Consider Proe that slope of string at any point x is equal in magnitude ratio of \chunk speed" (transerse elocity) to wae speed to f(z) f(x,t) where z x, t, n slope of f(x, t) is: Gien aboe wehae that From and we recognize that, desired. as we let z (x,t) and y f(x, t) f(z). Again.8.6.4 along a string. at that point. - -5 5 x (m) Use a similar argument to show that y(x; t) satises wae equation, y y for any (reasonable) function f. () : Similarly, we can write: (,): u, where u is chunk speed. Thus, we hae,u.8.6.4-4 - 4 t (s) f ( ) ( ) ( ) We follow similar steps for time deriaties: (,) f : f ( ) (, ) (, ) f : 4
since f f, refore, Howeer, Superposition 7. wae equation for transerse waes on a spring (or string) obeys Show that following ariation on wae equation obeys of superposition: principle sin(x) y We start with two solutions, y (x; t) and y (x; t), to wae equation aboe. Because y are both solutions: modied sin(x) y sin(x) y y see if y(x; t) Ay + By is also a solution. We'll start with left Let's of equation: side sin(x) A sin(x) y A y + B + Bsin(x) y Show that following ariation on wae equation does not obey principle of superposition: y sin y We'll try same method. If y (x; t) and y (x; t) are solutions, n y Asin sin + B y sin A y 6 B sin + B + f f : part of this equation is reponsible for it not obeying superposition? of superposition. That is, if y(x; t) and y(x; t) are solutions of principle wae equation, n y(x; t)+y(x; t) is a solution, too. y and y sin y Let's see if y(x; t) Ay + By is a solution. y [Ay + By] A y and sin(x) y does a sum of solutions fail to satisfy this wae equation? Because of Why sine, which is applied to y(x; t) (or more precisely, to its deriatie). that sine operation is not linear: sin( + sin( 6 sin(a +. [Ay + By] [Ay + By] So y(x; t) is a solution, too, and this wae equation obeys superposition. 5