Study sheet: Characterizing Wave Phenomena

📋 Course Outline

1. Sound Intensity and Attenuation
2. Diffraction Conditions and Measurement
3. Constructive and Destructive Interferences
4. Young’s Double-Slit Interference
5. Doppler Effect and Frequency Shift
6. Redshift in Astronomy

📖 1. Sound Intensity and Attenuation

🔑 Key Concepts & Definitions

• Sound intensity : the energy transported by a sound wave per unit time and per unit area, expressed in watts per square metre (W·m⁻²)

📝 Essential Points

📐 Formula — The sound intensity level is $L = 10\log\left(\frac{I}{I_0}\right)$ and is expressed in decibels (dB).

• The audibility threshold is $I_0 = 1.0 \times 10^{-12}\ \mathrm{W\,m^{-2}}$, whereas the pain threshold is 1 W·m⁻².

📌 Geometric attenuation occurs when increasing distance spreads the same power over larger spheres, whereas absorption attenuation occurs when a material between the source and receiver reduces the sound intensity.

💡 Memory Hook

Geometric attenuation depends on distance, whereas absorption depends on an intervening material.

📖 2. Diffraction Conditions and Measurement

🔑 Key Concepts & Definitions

• Diffraction : the spreading of a wave after it encounters an opening whose size is less than or equal to the wavelength

★ Must-know

📐 Formula — The characteristic diffraction angle satisfies $\theta = \frac{\lambda}{a}$, where θ is in radians, λ is the wavelength, and a is the opening size.

📌 Diffraction is more pronounced when the opening dimension a is close to or smaller than the wavelength λ.

Further detail

📐 Formula — For a central spot of width L on a screen at distance D, the opening size is $a = \frac{2\lambda D}{L}$ using the small-angle approximation.

• Diffraction occurs for both mechanical waves and light waves, so it demonstrates the wave nature of light.

💡 Memory Hook

A narrow opening relative to the wavelength causes the wave to spread.

📖 3. Constructive and Destructive Interferences

🔑 Key Concepts & Definitions

• Interference : the superposition of two waves that produces reinforcement in some regions and cancellation in others

★ Must-know

📐 Formula — Constructive interference occurs for a path difference $\delta = k\lambda$, whereas destructive interference occurs for $\delta = \left(k+\frac{1}{2}\right)\lambda$, with k an integer.

📌 Observable interference requires the two waves to have the same wavelength or frequency and to be synchronous, meaning that their phase difference remains constant.

📌 Constructive interference occurs when waves are in phase and their amplitudes add, whereas destructive interference occurs when waves are in opposite phase and their amplitudes cancel.

Further detail

• Noise-cancelling headphones detect ambient sound with an internal microphone and generate an opposite-phase signal that cancels the ambient noise.

💡 Memory Hook

In-phase waves reinforce; opposite-phase waves cancel.

📖 4. Young’s Double-Slit Interference

🔑 Key Concepts & Definitions

• Interfringe : the smallest distance between two consecutive points of constructive interference on the screen

📝 Essential Points

📐 Formula — The optical path difference at a screen point M is $\delta = \frac{e x}{D}$, where x is the point’s position and D is the slit-to-screen distance.

📐 Formula — The Young double-slit interfringe is $i = \frac{\lambda D}{e}$, where λ is the wavelength, D is the slit-to-screen distance, and e is the slit separation.

• In Young’s double-slit experiment, a laser illuminates two slits separated by a distance e, which act as two synchronous sources S₁ and S₂.

💡 Memory Hook

Path difference → screen position → interfringe.

📖 5. Doppler Effect and Frequency Shift

🔑 Key Concepts & Definitions

• Doppler effect : a change in the frequency of a wave perceived by an observer when the source moves relative to the observer

📝 Essential Points

📐 Formula — For a source approaching at speed v, the received frequency is $f_R = f_e\frac{c}{c-v}$, whereas for a receding source it is $f_R = f_e\frac{c}{c+v}$.

📐 Formula — The Doppler frequency shift is $\Delta f = f_R-f_e = f_e\frac{v}{c-v} \approx f_e\frac{v}{c}$ when v is much smaller than c.

📌 When the source approaches the observer, the perceived sound is higher and its frequency increases; when the source recedes, the perceived sound is lower and its frequency decreases.

💡 Memory Hook

Approaching source: higher perceived frequency; receding source: lower perceived frequency.

📖 6. Redshift in Astronomy

🔑 Key Concepts & Definitions

• Redshift : the shift of stellar spectral lines toward longer wavelengths when the star moves away from Earth

★ Must-know

📐 Formula — The wavelength and frequency of a wave are related by $\lambda = \frac{c}{f}$, so wavelength and frequency vary inversely.

Further detail

• Astronomers identify stellar motion by comparing the spectrum of light emitted by a star with the laboratory spectrum of the corresponding element.

💡 Memory Hook

A receding star produces longer wavelengths, causing a shift toward red.

📊 Synthesis Tables

Wave Phenomena Compared

Phenomenon	Condition	Observable effect
Diffraction	Opening size a ≤ wavelength λ	Wave spreading
Constructive interference	Path difference kλ	Wave reinforcement
Destructive interference	Path difference (k + 1/2)λ	Wave cancellation
Doppler effect	Source moves relative to observer	Perceived frequency shift