16.4
Graphing the Wave Function
Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From t…
Consider a sinusoidal wave traveling in the x-direction.
As its wave equation is the function of displacement and time, a particle's motion in the medium can be graphically represented by displacement-position and displacement-time graphs.
At a fixed time, the particle's displacement varies as a function of the position.
This represents the displacement of the particle from its equilibrium position. The wavelength can then be deduced from this graph.
Considering the case of a transverse wave on a string, the graph represents the actual shape of the string at a particular instant of time.
When a specific coordinate is chosen, graphing the wave equation results in a displacement-time graph.
Using this graph, the period—the time required for the wave to travel one wavelength—is deduced.
In the wave equation, the argument of the cosine function is called the phase of the wave.
The phase velocity is the speed at which the wave moves, keeping the phase constant.
Taking a derivative with respect to time, an expression for the phase velocity is obtained.
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Q1: What does a displacement versus position graph show for a wave?
A displacement versus position graph represents a snapshot of the wave at a fixed time, showing how particle displacement varies with position. For a transverse wave on a string, this graph depicts the actual shape of the string at that instant. The wavelength can be determined by measuring the distance between two consecutive crests or troughs with the same height and slope.
Q2: How do you find the period of a wave from a displacement-time graph?
A displacement-time graph shows particle displacement as a function of time at a specific position. The period is the time required for one complete oscillation of the particle. By observing the graph, you can identify the time interval between successive identical points in the wave cycle to determine the period.
Q3: What is the phase of a wave and what does it represent?
The phase is the argument of the cosine function in the wave equation, measured in radians. It determines which part of the sinusoidal cycle is occurring at a particular point and time. For a crest, the phase equals 0, 2π, 4π, etc.; for a trough, the phase equals π, 3π, 5π, etc.
Q4: What is phase velocity and how does it relate to wave motion?
Phase velocity is the speed at which a wave moves while keeping the phase constant. It describes how fast the wave pattern propagates through the medium. The phase velocity can be derived by taking the derivative of the wave equation with respect to time, providing a mathematical expression for wave propagation speed.
Q5: How are crests and troughs identified on a wave graph?
The crest is the highest point of the wave above equilibrium, while the trough is the lowest point below equilibrium. On a displacement versus position graph, crests appear as maximum positive displacements and troughs as maximum negative displacements. The distance between consecutive crests or troughs with identical height and slope equals the wavelength.
Q6: Why are two different graphs needed to fully describe a sinusoidal wave?
A sinusoidal wave depends on both position and time. The displacement versus position graph shows the wave's spatial shape at one instant, revealing wavelength. The displacement versus time graph shows how a single particle oscillates, revealing the period. Together, these graphs provide complete information about wave behavior in space and time.
Q7: How does the wave equation relate to graphical representations of waves?
The wave equation is a mathematical function of both position and time. By fixing time, you obtain a displacement versus position graph showing spatial wave structure. By fixing position, you obtain a displacement versus time graph showing temporal oscillation. These graphical representations make the wave equation's behavior visually interpretable for analyzing wave parameters.