Question

Standing waves are produced by the superposition of two waves $y_1=0.05\sin (3\pi t-2x)$ and $y_2=0.05\sin (3\pi t+2x)$ where $\mathrm{x}$ and $\mathrm{y}$ are in meters and $\mathrm{t}$ is in seconds. The speed (in $\mathrm{ms}-1$ ) of each wave is $\dots \dots$ (A) $1.5$ (B) $3.0$ (C) $3\pi /2$ (D) $3\pi$

Short answer

The speed of each wave is $v=\frac{3\pi }{2}$ m/s.

Step-by-step solution

Identify wavelength and frequency from the equations

We are given the two waves: $y_1=0.05\sin (3\pi t-2x)$ and $y_2=0.05\sin (3\pi t+2x)$. These are sinusoidal waves described by the form $y=A\sin (\omega t\pm kx)$, where A is the amplitude, ω is the angular frequency, k is the wave number, and ± depends on the direction of each wave. For both waves, we are given $\omega =3\pi$, k = 2. Next, we will find the lambda (wavelength) and frequency (f).

Calculate wavelength

The wavelength (lambda) is related to the wave number as follows: $k=\frac{2\pi }{\lambda }$ We know k = 2, therefore the wavelength is: $\lambda =\frac{2\pi }{k}=\frac{2\pi }{2}=\pi$ meters.

Calculate frequency

Frequency is related to the angular frequency as follows: $f=\frac{\omega }{2\pi }$ We know $\omega =3\pi$, and therefore the frequency is: $f=\frac{3\pi }{2\pi }=\frac{3}{2}$ Hz.

Determine wave speed

Now we have the wavelength ($\lambda =\pi$) and the frequency ($f=\frac{3}{2}$). We will use the wave speed equation ($v=\lambda f$) to calculate the speed of the wave: $v=(\pi )\left(\frac{3}{2}\right)=\frac{3\pi }{2}$ meters per second. The answer is choice (C): $v=\frac{3\pi }{2}$ m/s.

Key concepts

Wave Superposition

Wave superposition is a fascinating concept in physics, particularly when it comes to standing waves. A standing wave appears to be stationary, but it forms due to the superposition, or overlay, of two traveling waves moving in opposite directions. In our case, we have two waves, $y_1$ and $y_2$, given by the functions:

• $y_1=0.05\sin (3\pi t-2x)$
• $y_2=0.05\sin (3\pi t+2x)$

These waves have the same amplitude and angular frequency but travel in opposite directions, thus creating a standing wave. When waves superpose, their displacements either add up or cancel each other out, leading to constructive or destructive interference respectively.
Constructive interference occurs when the wave peaks align, amplifying the resultant wave. Destructive interference results when a wave peak aligns with a trough, effectively cancelling it out. This complex dancing creates the patterns of nodes and antinodes characteristic of standing waves.
Understanding this principle is crucial for predicting the behavior of waves in various mediums.

Wave Speed Calculation

Calculating wave speed is an integral part of understanding wave dynamics. The speed of a wave ($v$) can be calculated using the relationship between wavelength ($\lambda$) and frequency ($f$):$$v=\lambda f$$In the context of our standing waves, we've already determined the wavelength and frequency.
By plugging those values into our wave speed formula, we are able to determine the speed without guessing. Ensuring accuracy with this formula is essential in physics, as it relates to understanding the movement of waves through space and time. For our example, the calculation resulted in the speed of $v=\frac{3\pi }{2}{\text{ ms}}^{-1}$, indicating the choice of $C$ in the exercise.

Angular Frequency

Angular frequency ($\omega$) plays a pivotal role in wave dynamics, serving as a measure of how quickly a wave oscillates. It is comparable to the concept of ordinary frequency but is expressed in radians per second, which connects it to angular motion. The formula that relates angular frequency to frequency is:$$\omega =2\pi f$$Given the originally provided wave functions, $\omega$ for each wave was directly indicated as $3\pi$. This simplicity allowed us to swiftly determine values for other properties of the waves, such as frequency and wavelength.
Comprehending angular frequency enhances one's grasp of oscillatory motion, giving more insight into how waves propagate. It sheds light on the cycle time of one oscillation and connects to the wave's energy and speed, offering a more rounded understanding of wave phenomena.

Wavelength and Frequency Relationship

The relationship between wavelength ($\lambda$) and frequency ($f$) is crucial for analyzing wave behaviors. They are inversely proportional, meaning that as the wavelength increases, the frequency decreases, and vice versa. This relationship is a cornerstone of wave physics and is expressed through the wave speed equation:$$v=\lambda f$$In our problem, we successfully calculated the wavelength as $\pi$ meters and the frequency as $\frac{3}{2}$ Hz. With these values, we demonstrated how changes in one component, such as frequency, affect the other, such as wavelength.
Understanding this relationship helps explain many wave phenomena, from the pitch of sound to the color of light. It sets the foundation for more advanced exploration into wave mechanics and technology applications, making it a key concept for learners at all levels.