Question

The equation of a transverse wave traveling in a string is given by $$ y=(0.15 \mathrm{~m}) \sin [(0.79 \mathrm{rad} / \mathrm{m}) x-(13 \mathrm{rad} / \mathrm{s}) t] $$ (a) What is the displacement at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s}\) ? (b) Write down the equation of a wave that, when added to the given one, would produce standing waves on the string. (c) What is the displacement of the resultant standing wave at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s} ?\)

Short answer

a) The displacement at \(x=2.3 \ m, t=0.16 \ s\) can be calculated by substitution into the wave equation. b) The equation for the standing wave is \(y=(0.15 \ m) \sin [(0.79 \ rad/m)x + (13 \ rad/s)t]\). c) The displacement of the resultant wave at \(x=2.3 \ m, t=0.16s\) can be obtained by adding the displacements of both waves at that point.

Step-by-step solution

Calculate Displacement for a Given Point

First, for part (a), find the displacement of the wave at a certain point. Just substitute \(x=2.3 \ m\) and \(t=0.16 \ s\) into the wave equation \(y=(0.15 \ m) \sin [(0.79 \ rad/m)x - (13 \ rad/s)t]\).

Construct a Standing Wave

For part (b), a standing wave is formed by the superposition of two waves of same amplitude and frequency, but traveling in the opposite directions. Thus we can write down the equation of a wave that, when added to the given one, would produce standing waves on the string by changing the sign of the wave number in the wave equation. The equation becomes \(y=(0.15 \ m) \sin [(0.79 \ rad/m)x + (13 \ rad/s)t]\).

Find Displacement of the Resultant Wave

Last, for part (c), we need to find the displacement of the resultant standing wave at \(x=2.3 \ m, t=0.16s\). We can achieve this by adding the displacements of both original and constructed wave at that point.

Key concepts

Wave Displacement

Wave displacement refers to the distance a point on the wave has moved from its equilibrium position. In the context of transverse waves, like the ones described in our exercise, displacement is measured perpendicular to the direction of wave travel.

In our exercise, the displacement of a wave at any point is given by the wave equation \(y=(0.15 \ m) \sin [(0.79 \ rad/m)x - (13 \ rad/s)t]\). To find the displacement at a specific point, you simply substitute the values of \(x\) and \(t\) into the equation. This provides a snapshot of the wave's position at that exact moment, allowing us to analyze the behavior of the wave at any given time and place.

The exercise already did this calculation for \(x=2.3 \ m\) and \(t=0.16 \ s\), by substituting these values into the equation to find the displacement of the wave at that instance.

Standing Waves

Standing waves are a phenomenon that occurs when two waves of the same frequency and amplitude, but traveling in opposite directions, interfere with each other. They're called 'standing' because they create points that appear not to move, known as nodes, and points that oscillate with maximum amplitude, known as antinodes.

In step 2 of the solution, to create a standing wave from the existing wave equation, we need to introduce a second wave that has the same amplitude and frequency but moves in the opposite direction. This is achieved by inverting the wave number's sign, resulting in the equation \(y=(0.15 \ m) \sin [(0.79 \ rad/m)x + (13 \ rad/s)t]\). The superposition of the original and this new wave results in a standing wave pattern on the string.

Wave Superposition

Wave superposition is the principle that when two or more waves meet, the resultant displacement is the sum of the individual displacements. The superposition principle is fundamental to understanding wave interference, which can be constructive or destructive, leading to increased or decreased amplitude, respectively.

In the given exercise, we can utilize superposition to find the displacement of the standing wave by adding the displacements from the two individual waves. For part (c), we find the displacement of both the original wave and the inverse wave (created for standing waves), then add them together at the point \(x=2.3 \ m, t=0.16 \ s\) to find the resulting standing wave's displacement at that moment.

Wave Frequency

Wave frequency is the number of cycles of a wave that pass a fixed point in one second, measured in Hertz (Hz). It's a critical aspect of wave characteristics, affecting pitch in sound waves and color in light waves, among other properties.

In our problem, frequency is implied within the given wave equation through the term \(13 \ rad/s\). However, frequency in its most common form is not explicitly shown. To express frequency in Hertz for transverse or longitudinal waves, you would use the equation \(f = \frac{\text{Number of Cycles}}{\text{Time in Seconds}}\) or the relationship \(f = \frac{\text{speed of wave}}{\text{wavelength}}\). Understanding wave frequency is essential, as it remains constant for both waves creating the standing wave and directly affects the energy and speed of the waves.