Superposition and Standing Waves A standing wave occurs...

Superposition and Standing Waves A standing wave occurs when two travelling waves with the same wavelength and wave speed pass through each other in opposite directions. Mathematically the first wave is written as: Y1(x, t) = A cos(kx – ?t) and the second wave: Y2(x, t) = A cos(kx + ?t) where the wave number, k = 2?/?, the natural frequency ? = 2?/T, and ? and T are the wavelength and period of the waves, respectively. The positive and negative signs of ? show that the waves are travelling in opposite directions. Demonstrate that the sum of these two waves, Ysum = Y1 + Y2, results in a sinusoidal wave, Y(x) with a wavelength ? and an amplitude that is twice that of each wave individually. Furthermore, as time advances, the amplitude is oscillating between +A and -A with a period, T. Repeat the summation, only now assume the other harmonic wave form: Y1(x, t) = A sin(kx – ?t) and the second wave: Y2(x, t) = A sin(kx + ?t) The following trigonometric identities may be helpful: sin(? + ?) = sin ? cos ? + cos ? sin ? cos(? + ?) = sin ? cos ? – cos ? sin ? sin(-?) = – sin ? cos(-?) = cos ?

Transcript

00:01 Hi, so i have this question that tells me to prove that when i have a forward traveling wave and a backward traveling wave with the same frequency and the same wave number, and i sum them to give a standing wave, that the amplitude of the standing wave is twice the amplitude of either the forward or backward traveling waves.

00:27 For each of these waves, i was given these wave functions.

00:31 For the forward traveling wave we are told that y1 is equal a cost kx minus omega t and for the backward traveling wave y2 is equal to a cost kx plus omega t k is the wave number omega is the frequency now for us to get the standing wave corresponding to these two identical waves but with opposing directions, we have to sum the both of them.

01:06 And so y1 plus y2.

01:09 Because a is common, i will write a out there and i simply have the sum of these two cosine functions.

01:25 Now, to sum these cosine functions, i will have to use trigonometric identities.

01:32 And if we remember these identities for cosine functions, we have this that costs a plus b.

01:39 If we have this, it's a plus b.

01:40 If we equal to cos a, cost b minus sign a, sign b.

01:50 And that's cost a minus b is equal cost a, cos b, plus sign a, sign b.

02:04 Now if we use this information or the knowledge of these identities to attempt to sum these two waves, we will simply get this answer here again we'll write a out there and we can see that the back the forward traveling wave corresponds to the second identity for cos a minus b and so i will simply have cost k x cost omega t plus sign k x sign omega t and then i have the plus sign there and i write the identity for the backward traveling wave which is this top trigonometric identity and that gives me cos kx cos omega t minus sign k x sine x sine omega t we can see that these two functions are identical but with opposing signs and so i can cancel them again these two functions are the same and they have the same sign so when i sum them i have two as a coefficient and so what do we have we see that y1 plus y2 is equal to 2a cost k x cost omega t we've simply proven that the amplitude of the standing wave formed by the summation of the forward traveling wave and the backward traveling wave is twice the amplitude of either wave i will go ahead and solve the second part which has us to repeat the same thing but if the function describing these waves is now a sign function...

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