Problem 64 - Fundamentals of Physics

Question

The equation of a transverse wave traveling along a string is $$ y=0.15 \sin (0.79 x-13 t) $$ in which \(x\) and \(y\) are in meters and \(t\) is in seconds. (a) What is the displacement \(y\) at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s} ?\) A second wave is to be added to the first wave to produce standing waves on the string. If the second wave is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (b) \(y_{m},(\mathrm{c})\) \(k\), (d) \(\omega\), and (e) the correct choice of sign in front of \(\omega\) for this second wave? (f) What is the displacement of the resultant standing wave at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s}\) ?

Step-by-Step Solution

Answer

(a) -0.039 m, (b) 0.15 m, (c) 0.79 rad/m, (d) 13 rad/s, (e) negative sign, (f) -0.144 m.

Step 1: Substitute Values in the Wave Equation

The wave equation is given by \( y = 0.15 \sin (0.79 x - 13 t) \). We substitute \( x = 2.3 \) meters and \( t = 0.16 \) seconds.Calculate:\( y = 0.15 \sin (0.79 \times 2.3 - 13 \times 0.16) \).This simplifies to \( y = 0.15 \sin (1.817 - 2.08) \).Thus, \( y = 0.15 \sin (-0.263) \).

Step 2: Evaluate the Sine Function

Calculate the sine of the angle obtained.\( \sin (-0.263) \approx -0.260 \).Thus, \( y \approx 0.15 \times (-0.260) = -0.039 \) meters.

Step 3: Determine Amplitude of Second Wave

For standing waves, both waves must have the same amplitude. Therefore, \( y_m = 0.15 \) meters, the same as that of the first wave.

Step 4: Determine the Wave Number \( k \)

The wave number \( k \) from the given wave is already 0.79 rad/m, as we can see from \( 0.79x \) in the wave equation. Therefore, \( k = 0.79 \) rad/m for the second wave.

Step 5: Determine the Angular Frequency \( \omega \)

The angular frequency \( \omega \) is taken from the first wave's term \(-13 t\). Therefore, \( \omega = 13 \) rad/s for the second wave.

Step 6: Choose the Correct Sign for Standing Waves

For the waves to form standing waves, the second wave should have a negative sign, i.e., the wave is reflected. So the wave should be \( y_m \sin(kx - \omega t) \).

Step 7: Calculate Displacement of Standing Wave

The expression for the resultant standing wave is \( y = 2y_m \sin(kx) \cos(\omega t)\).Substitute \( y_m = 0.15 \), \( k = 0.79 \), \( \omega = 13 \):\( y = 2 \times 0.15 \sin(0.79 \times 2.3) \cos(13 \times 0.16) \).Evaluate: \( \sin(1.817) \approx 0.969 \) and \( \cos(2.08) \approx -0.495 \).Thus, \( y = 0.3 \times 0.969 \times (-0.495) \approx -0.144 \) meters.

Key Concepts

Transverse Waves

Transverse waves are a type of wave where the disturbance or oscillation occurs perpendicular to the direction of the wave's travel. A common example of transverse waves are ripples on a water surface. In the context of wave equations, such as the one given in the exercise, transverse waves are represented mathematically to show how the wave travels through a medium like a string or surface.

The essential characteristics of transverse waves include:

• Amplitude: The maximum displacement of points on the wave from its rest position.
• Wavelength: The distance over which the wave's shape repeats, typically measured from crest to crest.
• Frequency: The number of times the wave's cycle is completed in a given unit of time.
• Speed: How quickly the wave travels through the medium.

Understanding these properties helps us describe how energy is transmitted through the medium via the wave.

Standing Waves

Standing waves are formed when two waves of the same frequency and amplitude travel through a medium in opposite directions. These waves interfere with each other to produce a pattern that appears to be standing still. This phenomenon can occur in strings, air columns, and other restricted mediums where waves are reflected back upon themselves.

Key points about standing waves include:

• Nodes: Points of zero displacement where the waves always cancel each other out.
• Antinodes: Points of maximum displacement where the wave amplitudes add up.
• Formation: Typically occurs in confined spaces where waves reflect off boundaries.
• Equation: The standing wave can be modeled by an equation that combines both contributing wave equations and results in the form: \( y = 2y_m \sin(kx)\cos(\omega t) \).

Recognizing and understanding standing waves is vital in acoustics, musical instruments, and physics experiments.

Wave Displacement

Wave displacement refers to how far a point on the wave is from its original rest position at any given moment. It's a dynamic property that changes as the wave moves. In a transverse wave equation like the one given, displacement is often shown as a function of both position \( x \) and time \( t \).

Factors affecting wave displacement include:

• The amplitude of the wave, which dictates the maximum displacement possible.
• The position \( x \) along the wave and time \( t \), as waves change position and time affects the wave phase.
• The equation \( y = A \sin(kx - \omega t) \), where \( y \) gives the instantaneous displacement of the wave.

Accurate calculation of wave displacement is crucial for predicting wave behavior and interactions, especially in complex systems involving multiple waves.

Wave Number

The wave number \( k \) is a measure of the spatial frequency of a wave, which tells us how many wavelengths fit into a given unit of space. It's a crucial part of wave equations and is tied to the wave's wavelength \( \lambda \) by the relation \( k = \frac{2\pi}{\lambda} \).

Key aspects of wave number:

• Units: The wave number is measured in radians per meter (rad/m).
• Higher \( k \) values indicate shorter wavelengths, meaning more wave crests in a given space.
• In the given wave equation \( y = A \sin(kx - \omega t) \), the term \( kx \) represents the influence of the wave number on wave displacement.
• Determining \( k \) from the wave equation helps us understand the physical properties of the wave concerning its medium and speed.

The wave number is integral in linking the wave's physical properties with its mathematical description.

Angular Frequency

Angular frequency \( \omega \) is a measurement of how rapidly the wave oscillates in time. It is related to the wave's frequency \( f \) by the formula \( \omega = 2\pi f \). Angular frequency helps describe how fast the cyclic repetition of the wave's pattern occurs.

Important points about angular frequency are:

• Units: Angular frequency is measured in radians per second (rad/s).
• A higher \( \omega \) value indicates a faster oscillation rate of the wave.
• In the wave equation \( y = A \sin(kx - \omega t) \), \( \omega t \) represents the temporal component's impact on the wave's phase.
• Knowing \( \omega \) allows us to determine the energy and dynamics of wave behavior over time.

Angular frequency provides insights into the wave's energy distribution across time, particularly crucial in understanding harmonic motions and time-dependent wave phenomena.