Question

Determine the amplitude of the resultant wave when two sinusoidal waves having the same frequency and traveling in the same direction are combined, if their amplitudes are \(3.20 \mathrm{~cm}\) and \(4.19 \mathrm{~cm}\) and they differ in phase by \(\pi / 2 \mathrm{rad}\).

Short answer

The amplitude of the resultant wave is approximately 5.24 cm.

Step-by-step solution

Understanding the concept

When two waves of the same frequency and traveling in the same direction combine, the resultant wave's amplitude depends on the given amplitudes of the two waves and the phase difference. The phase difference refers to the displacement or the 'lag' between the two waves. The phase difference is given in radians.

Using the formula

We can use the formula \(A = \sqrt{{A_1}^2 + {A_2}^2 + 2A_1A_2 \cos\delta}\) to find the amplitude of the resultant wave. Here, \(A_1 = 3.20~cm\) and \(A_2 = 4.19~cm\) are the given amplitudes of the two waves and \(\delta = \pi/2~rad\) is the phase difference. The term \(\cos\delta\) measures the effect of the phase difference on the resultant amplitude.

Substituting the values

By substituting the values into the formula, we have \(A = \sqrt{{(3.20)}^2 + {(4.19)}^2 + 2*3.20*4.19*\cos(\pi/2)}\). Since \(\cos(\pi/2) = 0\), this simplifies to \(A = \sqrt{{(3.20)}^2 + {(4.19)}^2}\).

Calculating the amplitude

Calculate the value under the square root to find the resultant amplitude. This value can be approximated on a calculator.

Key concepts

Wave Interference

Imagine two waves meeting as they travel through the same medium, like ripples crossing each other's paths on a calm pond. This interaction between waves is known as wave interference, a phenomenon that can lead to a variety of resultants depending on the phase and amplitude of the interfering waves.

Now, when these waves come together, they can either add to each other's amplitude (constructive interference) or subtract from one another (destructive interference). The result of this superposition is a new wave pattern with regions of increased, reduced, or unchanged amplitude. For students to visualize this, imagining overlapping wave peaks and troughs can be quite helpful.

Furthermore, the principle of superposition, which is at the heart of wave interference, tells us that the displacement caused by the resultant wave at any point is the algebraic sum of the displacements caused by the individual waves at that same point.

Sinusoidal Waves

The sine wave or sinusoidal wave is a smooth, periodic oscillation that is a key concept in the study of waves and vibrations. These waves can be found in various fields such as acoustics, electronics, and even ocean waves. Sinusoidal waves are described by smooth peaks and troughs, with regular intervals between them.

The mathematical representation of such a wave involves the sine function, which explains the periodic nature of these oscillations. Key characteristics of sinusoidal waves include amplitude (the height of the wave), wavelength (the distance between consecutive peaks or troughs), frequency (how often the wave cycles occur per unit time), and phase (describes the position within one cycle of a periodic waveform).

Amplitude and Wavelength

Understanding these parameters is crucial for students. Amplitude in simple terms is the maximum extent of a vibration, which shows the energy carried by the wave. Wavelength is fundamentally linked to frequency and the speed at which the wave is traveling; thus, these characteristics serve as the foundations for complex wave analyses.

Phase Difference

The concept of phase difference is a vital part of understanding wave interactions. Phase can be thought of as the 'time' aspect of a wave's cycle - essentially, where the wave is in its cycle at a specific moment. Phase difference then measures how 'out of sync' two waves are with each other.

A phase difference is typically expressed in radians or degrees, where 360 degrees or 2π radians represent a full cycle. A phase difference of 0 means the waves are perfectly in sync, while a phase difference of π radians (or 180 degrees) indicates the waves are perfectly out of sync, leading to one wave's peak aligning with the other's trough - a setup for destructive interference.

Students must grasp that the conceptually simple shift in phase can have significant effects on the resultant wave when two waves combine. Even if the waves have identical amplitudes and frequencies, the phase difference can entirely change the outcome of their interaction.

Wave Combination

In the context of physics, wave combination, or the superposition of waves, is a process where two or more waves overlap and combine to form a new wave. The precise outcome of this process is governed by the characteristics of the individual waves, including their amplitudes, frequencies, and phase differences.

The formula for calculating the amplitude of the resultant wave, as seen in the textbook solution, embodies the essence of wave combination. It incorporates the amplitudes of the individual waves and the phase difference between them, showcasing how synchronized (or not) the waves are at the moment of their interaction.

Constructive vs. Destructive Interference

Specifically, students should be aware that when waves combine, they can reinforce each other (constructive interference) or partially cancel each other out (destructive interference). The formula used in the previous exercise is a guide to quantifying the exact amplitude of a combined wave, allowing for clear and precise predictions of the resultant wave patterns.