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# General Physics II PHYS 408

UNH

GPA 3.64

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This 2 page Class Notes was uploaded by Cayla Kilback on Thursday October 29, 2015. The Class Notes belongs to PHYS 408 at University of New Hampshire taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/231704/phys-408-university-of-new-hampshire in Physics 2 at University of New Hampshire.

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Date Created: 10/29/15

Traveling Waves Part II Transver39Se Waves on STr ings Chapter 17 Sec 79 Ener and Power in a Transverse Wave When a transverse wave travels down a string no material actually moves in the direction of the wave the material moves transverse to the direction of movement of the wave So what does move down the string One answer is the shape of the wave which is correct With that shape however there is also a transfer of energy The energy is in the movement of the bits of string kinetic energy and the stretching of the string potential energy Note yes you can also set up a wave in a chain which does not stretch Then there is still a form of potential energy namely the displacement of the links against the tension of the chain The rate of energy transmitted down the string can be found by looking at a small element dm with mass see last lecture udx The kinetic energy of this element is found from the kinetic energy equation K 7mv2 where v is the transverse velocity see last lecture which is MW at equal to So we get 2 dszydx 11 Differentiating our equation for a wave yxt ym sinkx wt with respect to t and inserting the result into 11 we get 2 dK 7udx aym coskx at 12 We now differentiate both sides with respect to t the Walt term is then just equal to the velocity of the wave so we find dK i dt 2 This gives the rate of change of the kinetic energy at a particular time and a particular location What we are actually interested in is the average power rate of change of the energy owing down the string that is transmitted down the string To get this we average eq 13 The average of c0s2x over 27239 is just 12 So dK dt uvw2y2m cos2 kx wt 13 V0 2y2m 14 The other component of the power traveling down the string is in the potential energy This is harder to derive but it is reasonable to accept that averages out to the same at eq 14 just like for a pendulum where the average kinetic and potential energies were equal The total power owing down the string is then PM vw2y2m 15 Note that the average power transported depends on the velocity amplitude and the frequency of the wave and the density of the string Since v i it depends on the tension too We u could rewrite 15 as 2 2 PM m y m 16 The Principle of Superposition and Interference If two or more waves travel down the same string and the string behaves as an ideal string they do not interact with each other The resulting deformation of the string is just the sum of the deformations of each all of the individual waves This is the called superposition In equation form we would write y xt y1xty2xt 17 The two waves do not affect each other but they do affect the shape of the resulting wave traveling down the string This is called interference A special result is obtained when the two waves have the same amplitude wavelength and frequency but a different phase Using the formula for the sum of two sine functions we obtain a resulting equation y xt2ym cos sinkx wt 18 If the two waves perfectly add up producing a new amplitude that is twice that of the original waves they are said to have fully constructive interference Then 0 2723 If on the other hand they completely cancel each other out they are said to have fully destructive interference and 7r37r Anything in between would be called intermediate interference but that term is not used too frequently Trick question What happens to the transfer of energy when there are two waves on a string that are in fully destructive interference

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