Nodes of a Standing Wave (Cosine) Learning Goal: To understand the concept of no

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Nodes of a Standing Wave (Cosine) Learning Goal: To understand the concept of nodes of a standing wave.The nodes of a standing wave are points where the displacement of the wave is zero at all times. Nodes are important for matching boundary conditions, for example that the point at which a string is tied to a support has zero displacement at all times (i.e.. the point of attachment does not move). Consider a standing wave, where y represents the transverse displacement of a string that extends along the x direction. Here is a common mathematical form for such a wave: y(x,t) = Acos(kappax)sm(omegat), where A is the maximum transverse displacement of the string (the amplitude of the wave), which is assumed to be nonzero, kappa is the wavenumber, omega is the angular frequency of the wave, and t is time. Part B At time t = 0. what is the displacement of the string t/(x,0)? Express your answer in terms of A, k, and other previously introduced quantities. What is the displacement of the string as a function of x at time T/A where T is the period of oscillation of the string? Express the displacement in terms of A, x, and k only. That is, evaluate omega. T/4 and substitute it in the equation for y(x,t). At which three points x1, x2. and X3 closest to x = 0 but with x greater than 0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points. Express the first three nonzero nodal points as multiples of the wavelength lambda, using constants like pi. List the factors that multiply lambda in increasing order, separated by commas.

Explanation / Answer

We will try to make the whole exercise, the given expression

Y(x, t)= A Cos(kx) Sin (wt)

Part B

The null desplacement for t=0

Sin (0)= 0

Y(x,t) = 0

Part C

calculate for t= T/4

. w =2pi/T pi symbolo for 3.14159

. w t = 2pi/T ( T/4) wt = 2pi/4 w t = pi/2

Sin ( pi/2) = 1

substituting in the initial expression

Y(x,t) = A Cos (k x)

Part D

nodes point

Y=0

Cos (k x) =0 kx = pi/2, 3pi/2, 5pi/2

L symbol for wavelenght

. k = 2pi/L (2pi/L) x = pi/2, 3pi/2, 5pi/2 x = (pi/2, 3pi/2, 5pi/2) L/2pi

. x = ¼ L, ¾ L, 5/4 L

For these points the oscilating Y is always zero regardless of the time

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