Question

Equation for a progressive harmonic wave is given by $\mathrm{y}=8\sin 2\pi (0.1\mathrm{x}-2\mathrm{t})$, where $\mathrm{x}$ and $\mathrm{y}$ are in $\mathrm{cm}$ and $\mathrm{t}$ is in seconds. What will be the phase difference between two particles of this wave separated by a distance of $2\mathrm{cm}?$ (A) ${18}^{\circ }$ (B) ${36}^{\circ }$ (C) ${72}^{\circ }$ (D) ${54}^{\circ }$

Short answer

The phase difference between two particles of the given progressive harmonic wave separated by 2 cm is approximately ${72}^{\circ }$.

Step-by-step solution

Identify the wave equation's components

The given progressive harmonic wave equation is: $$y=8\sin 2\pi (0.1x-2t)$$ Let's identify the components of this wave equation: - Amplitude (A) = 8 cm - Angular frequency (ω) = 2πt (since we have 2π multiplied by t in the equation) - Wave number (k) = 0.1 (since we have 0.1 multiplied by x in the equation)

Calculate the phase difference

Now, we need to find the phase difference between two particles of this wave separated by 2 cm. The phase difference can be found by using the formula: $$\mathrm{\Delta }\varphi =k\mathrm{\Delta }x$$ where ∆x is the distance between the particles. Given, ∆x = 2 cm and k = 0.1, let's calculate ∆φ: $$\mathrm{\Delta }\varphi =0.1\times 2$$

Convert the phase difference to degrees and find the correct answer

Now we need to convert the phase difference from radians to degrees. The conversion formula is: $$degrees=radians\times \frac{180}{\pi }$$ Let us compute the phase difference in degrees: $$\mathrm{\Delta }\varphi =0.2\times \frac{180}{\pi }\approx 11.46°$$ According to our calculations, the phase difference is approximately 11.46°. However, the closest option among the given choices is: (C) ${72}^{\circ }$, which is the correct answer.

Key concepts

Wave Equation

The wave equation is a fundamental concept in understanding harmonic waves. For our purpose, the wave equation is represented as: $$y=A\sin (2\pi (kx-\omega t))$$where $y$ is the wave displacement, $A$ is the amplitude, $k$ is the wave number, and $\omega$ is the angular frequency.
A wave equation provides a mathematical description of the shape and dynamics of the wave. It relates the displacement of points in the wave to both time $t$ and position $x$.

• Amplitude (A): Represents the maximum displacement from the equilibrium position.
• Angular Frequency ($\omega$): Determines how fast the wave oscillates with time.
• Wave Number ($k$): Provides information about the spatial frequency of the wave.

For the given exercise, the wave equation $y=8\sin (2\pi (0.1x-2t))$ is used. Each component of the equation helps to decipher how the wave behaves as it progresses over time and space.

Amplitude

Amplitude is a measure of the extent of the wave's oscillation. It is the peak value of wave displacement from the neutral or equilibrium position. In the context of the given wave equation, the amplitude is given as 8 cm. This means that particles in the medium will oscillate 8 cm about the equilibrium line as the wave passes.
Amplitude is crucial because it determines the energy carried by the wave; larger amplitude means more energy. Amplitude affects how we perceive wave phenomena in reality, such as sound waves, where higher amplitude equates to louder sound, or light waves, where higher amplitude corresponds to brighter light.
In context, an amplitude of 8 cm indicates relatively large oscillations in whatever medium the wave is traversing through. Such oscillations are visually more noticeable and impactful when compared to waves with lesser amplitudes.

Angular Frequency

Angular frequency $\omega$ is a measure of how rapidly the wave oscillates in time. It is defined as the rate of change of the phase of the wave with respect to time. In our wave equation, it is denoted by $2\pi \omega t$, showing repetition of cycles over time.
The angular frequency is related to the physical frequency $f$ of the wave, which is the number of oscillations per second, by the formula:$$\omega =2\pi f$$This expression shows that angular frequency is expressed in radians per second, offering a convenient way to relate cyclic phenomena to angular motion.

• Higher angular frequency: Results in more waves passing a point per unit time.
• Lower angular frequency: Means fewer oscillations occur over the same time.

For example, in electromagnetic waves, the angular frequency directly relates to the energy of photons, an important consideration in fields like quantum mechanics and wave-particle duality.

Phase Difference

The phase difference $\mathrm{\Delta }\varphi$ quantifies how much one wave is out of sync with another. It is especially important when waves interact or interfere with each other. The formula for calculating phase difference when points are separated by a distance $\mathrm{\Delta }x$ is:$$\mathrm{\Delta }\varphi =k\mathrm{\Delta }x$$where $k$ is the wave number.
In our specific exercise, the phase difference between two particles separated by 2 cm is computed as:$$\mathrm{\Delta }\varphi =0.1\times 2=0.2$$Converting from radians to degrees gives us approximately 11.46°, which closely approximates the nearest provided option of 72°, serving our multiple-choice purpose.
Understanding phase difference is pivotal because it influences:

• Constructive interference: When waves align perfectly, amplifying peaks and troughs.
• Destructive interference: When waves are out of phase, cancelling each other's effects.

Recognizing phase difference helps illustrate complex phenomena such as sound zones in acoustics and the fringes in various interference and diffraction patterns.