Understanding and Using Displacement, Amplitude, Phase Difference, Period, Frequency, and Wavelength in Progressive Waves

Introduction

Key Concepts

Displacement

Displacement in wave mechanics refers to the distance a point on the medium moves from its equilibrium position as the wave passes through. It is a vector quantity, having both magnitude and direction. Displacement can be positive or negative, indicating the direction of movement relative to the equilibrium.

Mathematically, displacement ($y$) at any point and time can be expressed as: $$ y(x, t) = y_m \sin(kx - \omega t + \phi) $$ where $y_m$ is the maximum displacement, $k$ is the wave number, $\omega$ is the angular frequency, and $\phi$ is the phase constant.

**Example:** In a transverse wave on a string, if a point on the string moves 5 cm above its rest position, the displacement at that point is +5 cm.

Amplitude

Amplitude is the maximum displacement of points on a wave from their equilibrium positions. It is a measure of the wave's strength or intensity. In mechanical waves, a larger amplitude means a wave carries more energy.

The amplitude ($A$) is represented in the wave equation as: $$ y(x, t) = A \sin(kx - \omega t + \phi) $$

**Example:** A loud sound wave has a large amplitude, whereas a faint sound wave has a small amplitude.

Phase Difference

Phase difference refers to the difference in phase between two points in a wave or between two waves. It is usually measured in degrees or radians. Phase difference can cause constructive or destructive interference when waves overlap.

If two waves have equations: $$ y_1 = A \sin(kx - \omega t) $$ $$ y_2 = A \sin(kx - \omega t + \delta) $$ the phase difference ($\delta$) is the angle by which one wave is ahead or behind the other.

**Example:** If $\delta = 0$, the waves are in phase, leading to constructive interference. If $\delta = \pi$, the waves are out of phase, resulting in destructive interference.

Period

The period ($T$) of a wave is the time taken for one complete cycle of the wave to pass a given point. It is the reciprocal of the frequency.

$$ T = \frac{1}{f} $$ where $f$ is the frequency.

**Example:** If a wave has a frequency of 50 Hz, its period is $T = \frac{1}{50} = 0.02$ seconds.

Frequency

Frequency ($f$) is the number of complete waves that pass a given point in one second. It is measured in hertz (Hz).

The relationship between frequency and period is given by: $$ f = \frac{1}{T} $$

**Example:** A guitar string vibrating at 440 Hz produces the musical note A4.

Wavelength

Wavelength ($\lambda$) is the distance between successive crests or troughs of a wave. It is a key factor in determining the wave's behavior and interactions with materials.

Wavelength is related to wave speed ($v$) and frequency by the equation: $$ \lambda = \frac{v}{f} $$

**Example:** Light waves have wavelengths ranging from about 400 nm (violet) to 700 nm (red).

Advanced Concepts

In-Depth Theoretical Explanations

Building upon the foundational concepts, it is essential to explore the mathematical derivations and interrelationships between these wave properties. Starting with the wave equation: $$ y(x, t) = y_m \sin(kx - \omega t + \phi) $$ where:

• $y_m$ = amplitude
• $k$ = wave number ($k = \frac{2\pi}{\lambda}$)
• $\omega$ = angular frequency ($\omega = 2\pi f$)
• $\phi$ = phase constant

By substituting the expressions for $k$ and $\omega$, we can derive relationships such as: $$ y(x, t) = y_m \sin\left(\frac{2\pi}{\lambda}x - 2\pi f t + \phi\right) $$ Simplifying, we obtain a clearer view of how wavelength and frequency influence the wave's behavior.

**Derivation Example:** To derive the relationship between wave speed, frequency, and wavelength: Starting with: $$ v = \lambda f $$ This equation indicates that the wave speed is the product of its wavelength and frequency, fundamental in fields like optics and acoustics.

Complex Problem-Solving

Consider a wave traveling along a string with a wavelength of 2 meters and a frequency of 5 Hz. Determine the wave speed and the displacement at a point 1 meter from the origin at $t = 0.1$ seconds.

**Solution:**

• Wave speed ($v$): $$ v = \lambda f = 2 \times 5 = 10 \text{ m/s} $$
• Displacement ($y$) using the wave equation with $\phi = 0$: $$ y(x, t) = y_m \sin\left(\frac{2\pi}{\lambda}x - 2\pi f t\right) $$ Assuming $y_m = 1$ cm for simplicity: $$ y(1, 0.1) = 1 \sin\left(\frac{2\pi}{2} \times 1 - 2\pi \times 5 \times 0.1\right) = 1 \sin(\pi - \pi) = 1 \times 0 = 0 \text{ cm} $$

**Advanced Example:** Analyze the interference pattern formed by two waves of equal amplitude and frequency traveling in the same medium but with a phase difference of $\frac{\pi}{2}$. Determine the resultant amplitude.

**Solution:**

• Using the principle of superposition: $$ y_{\text{total}} = y_1 + y_2 = A \sin(\omega t) + A \sin\left(\omega t + \frac{\pi}{2}\right) = A \sin(\omega t) + A \cos(\omega t) $$
• The resultant amplitude ($A_{\text{resultant}}$) is: $$ A_{\text{resultant}} = \sqrt{A^2 + A^2} = A\sqrt{2} $$

Interdisciplinary Connections

Wave principles are not confined to physics but extend to various fields such as engineering, music, and medicine. For instance:

• Engineering: Understanding wave displacement and amplitude is crucial in designing structures that can withstand seismic waves during earthquakes.
• Music: The frequency and amplitude of sound waves determine pitch and loudness, which are fundamental in musical acoustics.
• Medicine: Ultrasound imaging relies on high-frequency sound waves, utilizing phase difference and wavelength to create detailed images of the body's interior.

Furthermore, electromagnetic waves, governed by the same principles, are pivotal in telecommunications, optics, and even quantum physics, showcasing the pervasive nature of wave concepts across scientific disciplines.

Comparison Table

Summary and Key Takeaways