Problem 82 - Fundamentals of Physics

Question

Two sinusoidal waves of the same wavelength travel in the same direction along a stretched string. For wave \(1, y_{m}=3.0 \mathrm{~mm}\) and \(\phi=\) \(0 ;\) for wave \(2, y_{m}=5.0 \mathrm{~mm}\) and \(\phi=70^{\circ} .\) What are the (a) amplitude and (b) phase constant of the resultant wave?

Step-by-Step Solution

Answer

The amplitude is approximately 6.65 mm, and the phase constant is approximately 45°.

Step 1: Understand the given values

We have two sinusoidal waves: - Wave 1 with amplitude \( y_{m1}=3.0 \, \text{mm} \) and phase \(\phi_1=0\).- Wave 2 with amplitude \( y_{m2}=5.0 \, \text{mm} \) and phase \(\phi_2=70^{\circ}\).They travel in the same direction along a stretched string. We need to find the amplitude and phase constant of the resultant wave.

Step 2: Use the principle of superposition

According to the principle of superposition, the resultant wave is the sum of the two individual waves. For two waves \( y_1 = y_{m1} \sin(kx - \omega t + \phi_1) \) and \( y_2 = y_{m2} \sin(kx - \omega t + \phi_2) \), the resultant wave can be given as \( y = y_{m1} \sin(kx - \omega t) + y_{m2} \sin(kx - \omega t + 70^{\circ}) \).

Step 3: Calculate amplitude of resultant wave

The amplitude of the resultant wave is \[ y_m = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2}\cos(\phi_2 - \phi_1)} \]Substituting the given values:\[ y_m = \sqrt{3.0^2 + 5.0^2 + 2 \times 3.0 \times 5.0 \times \cos(70^{\circ})} \]Calculate the values:\( \cos(70^{\circ}) \approx 0.342 \).Substitute into the equation:\[ y_m = \sqrt{9 + 25 + 30 \times 0.342} \]\[ y_m = \sqrt{9 + 25 + 10.26} \]\[ y_m = \sqrt{44.26} \approx 6.65 \text{ mm} \].

Step 4: Calculate phase constant of resultant wave

The phase constant \( \phi \) of the resultant wave can be found using \[ \tan \phi = \frac{y_{m2} \sin \phi_2}{y_{m1} + y_{m2} \cos \phi_2} \]Substitute the given values:\[ \tan \phi = \frac{5.0 \times \sin 70^{\circ}}{3.0 + 5.0 \times \cos 70^{\circ}} \]Calculate the trigonometric values:\( \sin(70^{\circ}) \approx 0.94 \) and \( \cos(70^{\circ}) \approx 0.342 \).Substitute back:\[ \tan \phi = \frac{5.0 \times 0.94}{3.0 + 5.0 \times 0.342} \]\[ \tan \phi = \frac{4.7}{4.71} \]\[ \phi = \tan^{-1}(1.0) \approx 45^{\circ} \].

Key Concepts

Superposition Principle

The Superposition Principle is a fundamental concept in wave mechanics. It allows us to predict the behavior of two or more overlapping waves. When waves meet, they don't physically alter each other. Instead, they combine temporary based on their amplitudes and phases.
The principle states that the resultant displacement at any point is the sum of the displacements from each individual wave at that point. For instance, if two waves, each described by their sinusoidal functions, overlap, their combined effect is a new wave which is their mathematical sum.

• This principle helps in understanding complex wave phenomena.
• It's applicable in various scenarios like sound waves, light waves, and other electromagnetic waves.

In our exercise, this principle was used to combine two sinusoidal waves traveling along a string, resulting in a new wave with a unique amplitude and phase constant.

Wave Amplitude

Wave Amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. It's essentially the 'height' of the wave when represented on a graph.
In wave physics, amplitude is directly related to the energy carried by the wave. The larger the amplitude, the more energy the wave carries.

• Mathematically, amplitude is a crucial parameter in wave functions: it determines the peak value of oscillating quantities.
• In our problem, wave 1 has an amplitude of 3 mm, and wave 2 has 5 mm.

To find the amplitude of the resultant wave, we employ the formula: \[ y_m = \sqrt{y_{m1}^2 + y_{m2}^2 + 2y_{m1}y_{m2}\cos(\phi_2 - \phi_1)} \]This formula relates to how two wave amplitudes combine depending on their phase differences.

Phase Constant

The Phase Constant is a term that describes the initial angle of a wave when it begins. It's an essential component in wave equations, impacting where the wave starts on its journey.
Phase constant is crucial when multiple waves interfere, as it dictates how waves align when they meet.

• In the exercise at hand, wave 1 has a phase of 0°, while wave 2 has a phase of 70°.
• To find the resultant wave's phase, we use the formula: \[ \tan \phi = \frac{y_{m2} \sin \phi_2}{y_{m1} + y_{m2} \cos \phi_2} \]

Calculating this provides the phase shift necessary to describe the interference pattern of the combined waves, explaining concepts like constructive and destructive interference.

Trigonometry in Waves

Trigonometry is an indispensable tool in understanding wave behavior. Waves are periodic by nature and are often described using trigonometric functions like sine and cosine.
In our problem, trigonometric identities help in calculating combined wave properties, such as amplitude and phase. The sine and cosine of angles between waves play critical roles in determining how these waves interact.

• The cosine function helps to account for differences in phases between waves, especially for calculating amplitude.
• The tangent function is used to find the phase constant of the resultant wave from interfering waves.

Using trigonometry, we understand wave interactions better, from simple oscillations to complex interference patterns.