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Chapter 9: Problem 58

The equation \(y=a \sin \frac{2 \pi}{\lambda}(v t-x)\) is expression for (A) Stationary wave of single frequency along \(x\)-axis. (B) A simple harmonic motion. (C) A progressive wave of single frequency along \(x\)-axis. (D) The resultant of two SHMs of slightly different frequencies.

Short Answer

Expert verified
The given equation represents a progressive wave of single frequency along the x-axis. Therefore, the correct answer is (C) A progressive wave of single frequency along the x-axis.

Step by step solution

01

Identify the main components of the equation

The given equation is: y = a * sin ( (2蟺/位) * (vt - x) ) Here, y represents the wave function which gives the value of the y-coordinate at different points in xy-plane. The amplitude is represented by a, the wavelength is represented by 位, the speed of the wave is represented by v, the time is represented by t, and the position along the x-axis is represented by x. The sine function indicates that this is a sinusoidal wave.
02

Analyze which type of wave is represented by the equation

The given equation has both time (t) and position (x) in its argument. This indicates that it is a wave that varies with time and propagates along the x-axis. Since the sine function is present, we can see that it's a sinusoidal wave.
03

Determine whether the equation represents a stationary wave or a progressive wave

In a stationary wave, the position of the wave nodes and antinodes remains constant with time. The equation of a stationary wave would have the sum of two sinusoidal functions, one with positive x and the other with negative x, keeping the nodes always at the same position in time. In a progressive wave, the wave disturbance travels in a particular direction and the equation of a progressive wave consists of sinusoidal terms involving (vt 卤 x). The given equation has the form: y = a * sin ( (2蟺/位) * (vt - x) ) which has the form of a progressive wave. Therefore, the equation represents a progressive wave and not a stationary wave.
04

Check if the wave is a simple harmonic motion and the other options

A simple harmonic motion (Option B) is a wave that oscillates in time, but does not propagate in space. Since our given equation involves the variable x for the position along the x-axis, it is not a simple harmonic motion. Option D involves the resultant of two simple harmonic motions with slightly different frequencies. This would result in a beat phenomenon which is not represented by the given equation as it has only one sinusoidal function. Based on the analysis in the previous steps, we can see that the given equation represents a progressive wave of a single frequency along the x-axis. Therefore, the correct answer is: (C) A progressive wave of single frequency along the x-axis.

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