QUESTION Standing waves As shown in Figure 2, a standing wave is mains fixed in

Question

QUESTION

Standing waves As shown in Figure 2, a standing wave is mains fixed in position over time. Figure 2:Standing waves on a string. Standing waves can be produced by superposing two identical waves (having the same ampli- tude, speed, and wavelength) moving in opposite directions. In the lab, this is accomplished by fixing an end of string from which the incoming wave can almost perfectly reflect and superpose itself. The mathematical expression of a standing wave can be derived by adding together two counter-propagating waves y+ and y- (as shown in Section 3.1), as a wave whose shape re- /2 yswlx, t) = y+(x, t) + y.(x, t) = 2A cos(2nft) sinax)

Explanation / Answer

wave moving in positive direction

y+ = A sin(kx - 2*pi*f*t)

wave moving in negative direction

y- = A sin(kx + 2*pi*f*t)

resultant

y = (y+) + (y-)

= A [sin(kx)*cos(2*pi*f*t) - cos(kx) *sin(2*pi*f*t)] + A [sin(kx)*cos(2*pi*f*t) + cos(kx) *sin(2*pi*f*t)]

   = 2A*sin(kx)* cos(2*pi*f*t)

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