Question

A standing wave resulte from sum of two transverse travelling waves given by, y1=0.050cos(πx−4πt) and y2=0.050cos(πx+4πt)

, where x,y1,y2 are in metre and t in seconds. The smallest positive value of x that corresponds to a node is
A. 0 m
B. 0.5 m
C. 1.5 m
D. 1 m

Answer 1

The smallest positive value of x that corresponds to a node in the standing wave described is B. 0.5 m because this is where the cosine term becomes zero and thus the amplitude of the wave is zero.

Step-by-step explanation:

The student is asking about the conditions for the formation of a node in a standing wave resulting from the superposition of two traveling waves. Nodes are points in a standing wave where the amplitude is always zero. Given the equations for the two waves y1=0.050cos(πx-4πt) and y2=0.050cos(πx+4πt), we can find the resulting standing wave by adding these two equations:

Y(x,t) = y1 + y2 = 0.050cos(πx-4πt) + 0.050cos(πx+4πt)

Using the trigonometric identity for the sum of cosines, we get:

Y(x,t) = 2 × 0.050 × cos(πx) × cos(4πt)

To find the smallest positive value of x corresponding to a node, we need to find the values of x for which cos(πx)=0 because this will make the amplitude of the standing wave zero at that point. The smallest positive x for which cos(πx) is zero is when x is half the period of the cosine function, which is at x=0.5 meters.

So, the correct answer is B. 0.5 m.