A stationary wave is represented by the following expression:𝑦 = 5cos𝜋𝑥sin𝜋𝑡where y and x in centimeters and t in seconds. Determine:(a) the three smallest value of x (x >0) that corresponds toi. nodes
Question
A stationary wave is represented by the following expression:𝑦 = 5cos𝜋𝑥sin𝜋𝑡where y and x in centimeters and t in seconds. Determine:(a) the three smallest value of x (x >0) that corresponds toi. nodes
Solution
(a) Nodes in a stationary wave are points where the amplitude is always zero. This means that the displacement y is always zero at these points.
The equation of the wave is given as y = 5cos(πx)sin(πt).
For a node, y = 0. This means that either cos(πx) = 0 or sin(πt) = 0.
Since we are looking for the values of x, we set cos(πx) = 0.
The cosine function has a value of zero at odd multiples of π/2. Therefore, we can set πx equal to odd multiples of π/2 and solve for x.
i. For the first node (n=1), we have:
πx = (2*1 - 1) * π/2 πx = π/2 x = 1/2 cm
ii. For the second node (n=2), we have:
πx = (2*2 - 1) * π/2 πx = 3π/2 x = 3/2 cm
iii. For the third node (n=3), we have:
πx = (2*3 - 1) * π/2 πx = 5π/2 x = 5/2 cm
So, the three smallest values of x that correspond to nodes are x = 1/2 cm, x = 3/2 cm, and x = 5/2 cm.
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