Fundamentals of Waves, Light, and Sound

Understanding Wave Behavior in Different Media

Waves travel through a variety of media—air, water, solids—each of which influences the wave’s speed, wavelength, and ultimately the sound we hear. In this module we explore how a tuning fork’s pitch changes when the surrounding medium changes from air to water, and why the frequency remains constant while the wavelength adjusts.

Key Concept: Frequency vs. Wavelength

When a source vibrates, it produces a fixed frequency (the number of oscillations per second). The wave’s speed v in a medium is related to its frequency f and wavelength λ by the fundamental equation:

• v = f \times λ

Because the tuning fork’s vibration rate does not change when it is moved from air to water, the frequency f stays the same. However, the speed of sound is higher in water (≈1500 m/s) than in air (≈340 m/s). To satisfy the equation, the wavelength must become shorter in water.

Why Pitch Appears Lower in Water

Our ears interpret pitch primarily from frequency, but the human auditory system also responds to the wavelength of the sound wave as it reaches the ear. A shorter wavelength in a faster medium can cause the ear to perceive a slightly lower pitch, especially when the source is not directly audible (e.g., when the sound is transmitted through a solid container into water).

Think of a guitar string: tightening the string raises the frequency, producing a higher pitch. In the tuning‑fork scenario, the "tightening" is not of the string but of the medium—water lets the wave travel faster, forcing the wavelength to shrink while the frequency stays the same.

How Rainbows Form: Refraction, Internal Reflection, and Refraction Again

Rainbows are one of the most beautiful demonstrations of wave optics. When white sunlight enters a spherical raindrop, three distinct optical processes occur:

1. Refraction – the light bends as it passes from air into the denser water, separating into its constituent colors.
2. Internal reflection – the light reflects off the back surface of the droplet, staying inside the water.
3. Refraction again – the light exits the droplet, bending a second time and spreading the colors further.

This sequence—refraction, internal reflection, then refraction—creates the characteristic circular arc of a rainbow.

Why Each Step Matters

• First refraction begins the dispersion of colors because each wavelength bends at a slightly different angle.
• Internal reflection preserves the separated colors while redirecting the light toward the observer.
• Second refraction amplifies the angular separation, making the colors visible over a wide range of viewing angles.

Imagine a pool cue striking a cushion (first refraction), bouncing off the far side of the table (internal reflection), and then shooting out toward the edge (second refraction). The cue’s path mirrors the light’s journey inside the droplet.

Transverse Waves on a String: Tension, Speed, and Wavelength

When a transverse wave travels along a string, its speed depends on the tension T and the linear mass density μ of the string:

• v = \sqrt{T/μ}

Increasing the tension raises the wave speed. If the source continues to vibrate at the same frequency, the wavelength must increase to keep the relationship v = fλ valid.

Resulting Changes

With higher tension:

• Speed increases because the string is tighter.
• Wavelength increases proportionally, while the frequency stays constant.

This explains why the correct answer to the quiz question is that both wavelength and speed increase while frequency remains unchanged.

Frequency–Wavelength Relationship for Electromagnetic Waves

Electromagnetic (EM) waves, such as visible light, travel through vacuum at the constant speed c ≈ 3.00 × 10⁸ m/s. Their frequency f and wavelength λ are inversely related:

• c = fλ

Because c is fixed, a higher frequency necessarily means a shorter wavelength, and vice‑versa. This inverse relationship is fundamental to many phenomena, from radio broadcasting (low‑frequency, long‑wavelength) to X‑rays (high‑frequency, short‑wavelength).

Why the Relationship Holds

In a vacuum there are no material properties to alter the wave speed; the only way to satisfy the equation is for f and λ to adjust inversely. This principle underlies the color dispersion in prisms and the design of antennas.

Interference of Sound Waves in Different Media

Interference occurs when two or more waves overlap, producing regions of constructive (amplified) or destructive (reduced) amplitude. The key factor is the **path difference** between the waves, not the medium they travel through.

Scenario: Two Identical Speakers

One speaker emits sound into a solid block, the other into air. At a point equidistant from both sources, the waves arrive with the same phase if the path lengths are equal. However, because the speed of sound differs between air (≈340 m/s) and the solid (often >5000 m/s), the **time** taken to travel the same distance varies.

Despite the speed difference, the condition for destructive interference remains:

• Path difference = (n + ½) λ, where n is an integer.

Thus, if the geometric path difference equals half a wavelength, the waves cancel each other, producing a node of silence. This demonstrates that interference is possible across media, provided the waves meet the phase‑difference criterion.

Key Takeaway

The correct answer is that destructive interference can occur when the path difference equals half a wavelength, regardless of the differing propagation speeds in each medium.

Summary of Core Principles

• Wave speed, frequency, and wavelength are linked by v = fλ. Changing the medium alters speed, which in turn adjusts wavelength if frequency stays constant.
• Rainbow formation follows the sequence: refraction → internal reflection → refraction.
• Tension in a string raises both wave speed and wavelength while preserving frequency.
• Electromagnetic waves in vacuum obey c = fλ, making frequency and wavelength inversely proportional.
• Interference depends on path difference, not on the medium’s speed, allowing constructive or destructive patterns even when waves travel through different substances.

By mastering these relationships, students gain a solid foundation for tackling more advanced topics in acoustics, optics, and wave mechanics.