Two waves, y1=(2.50 mm) sin[(25.1 rad / m) x-(440 rad / s) t] and y2=(1.50 mm) sin[(25.1 rad / m

step 1 16 .92. So we have two waves traveling along a string. They're given here. You see that their KX

Two waves, y1=(2.50 mm) sin[(25.1 rad / m) x-(440 rad / s)...

Two waves, $$y_{1}=(2.50 \mathrm{mm}) \sin [(25.1 \mathrm{rad} / \mathrm{m}) x-(440 \mathrm{rad} / \mathrm{s}) t]$$ and $$y_{2}=(1.50 \mathrm{mm}) \sin [(25.1 \mathrm{rad} / \mathrm{m}) x+(440 \mathrm{rad} / \mathrm{s}) t]$$ travel along a stretched string. (a) Plot the resultant wave as a function of $t$ for $x=0, \lambda 8, \lambda / 4,3 \lambda / 8,$ and $\lambda / 2,$ where $\lambda$ is the wavelength. The graphs should extend from $t=0$ to a little over one period. (b) The resultant wave is the superposition of a stand- ing wave and a traveling wave. In which direction does the traveling wave move? (c) How can you change the original waves so the resultant wave is the superposition of standing and traveling waves with the same amplitudes as before but with the traveling wave moving in the opposite direction? Next, use your graphs to find the place at which the oscillation amplitude is (d) maximum and (e) minimum. (f) How is the maximum amplitude related to the amplitudes of the original two waves? (g) How is the minimum amplitude related to the amplitudes of the original two waves?

Two waves,
$$y_{1}=(2.50 \mathrm{mm}) \sin [(25.1 \mathrm{rad} / \mathrm{m}) x-(440 \mathrm{rad} / \mathrm{s}) t]$$ and $$y_{2}=(1.50 \mathrm{mm}) \sin [(25.1 \mathrm{rad} / \mathrm{m}) x+(440 \mathrm{rad} / \mathrm{s}) t]$$ travel along a stretched string. (a) Plot the resultant wave as
a function of $t$ for $x=0, \lambda 8, \lambda / 4,3 \lambda / 8,$ and $\lambda / 2,$ where $\lambda$ is the
wavelength. The graphs should extend from $t=0$ to a little over one period. (b) The resultant wave is the superposition of a stand-
ing wave and a traveling wave. In which direction does the traveling wave move? (c) How can you change the original waves so the resultant wave is the superposition of standing and traveling
waves with the same amplitudes as before but with the traveling
wave moving in the opposite direction? Next, use your graphs to find the place at which the oscillation amplitude is (d) maximum
and (e) minimum. (f) How is the maximum amplitude related to
the amplitudes of the original two waves? (g) How is the minimum
amplitude related to the amplitudes of the original two waves?

Key Concepts

Amplitude Relationships

The relationship between wave amplitudes in interference is central to predicting the resultant amplitude. In cases of constructive interference, the maximum amplitude can be the sum of the individual wave amplitudes, whereas destructive interference can lead to a minimum amplitude equivalent to the absolute difference of the individual amplitudes. This concept is essential for determining how the amplitudes combine in the problem, particularly in parts (f) and (g) where maximum and minimum amplitudes are analyzed.

Nodes and Antinodes

Nodes and antinodes are fundamental features of standing waves. Nodes are positions where the interfering waves cancel each other completely, resulting in zero amplitude, while antinodes are points of maximum amplitude due to constructive interference. Identifying these points in a superposed wave helps in understanding the spatial distribution of energy in a standing wave pattern.

Wave Interference

Wave interference involves the combination of two or more waveforms where their amplitudes add together, resulting in regions of constructive interference (increased amplitude) and destructive interference (reduced or zero amplitude). This concept underpins the formation of both the standing wave and the traveling wave components in the resultant wave.

Traveling Waves

Traveling waves are wave disturbances that move through a medium, carrying energy from one point to another. Their waveform moves in a specific direction, which is determined by the phase term in the wave equation. In the given context, one component of the resultant wave retains the characteristics of a traveling wave, and the direction of propagation can be deduced from the sign of the time-dependent phase.

Standing Waves

Standing waves are formed by the interference of two waves of the same frequency and amplitude traveling in opposite directions. They are characterized by stationary nodes, where the displacement is always zero, and antinodes, where the amplitude reaches a maximum. In the problem, part of the resultant wave exhibits standing wave behavior, identifiable by fixed positions of maximum and minimum amplitude.

Superposition Principle

The superposition principle states that when multiple waves traverse the same medium, their displacements add algebraically to form the net displacement. This principle is fundamental to understanding how the given wave functions combine to produce the resultant waveform, whether it be a pure traveling wave, a standing wave, or a combination of both.

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