Waves: Energy, Intensity, Loudness, and the Doppler Effect

Waves: Energy, Intensity, Loudness, and the Doppler Effect

Recap: The Wave Model

The wave model describes how disturbances propagate through space and time, transferring energy without transferring matter. Waves can be classified based on the direction of particle motion relative to the direction of wave propagation.

• Transverse Waves: Particles of the medium move perpendicular to the direction of wave travel (e.g., waves on a string).

• Longitudinal Waves: Particles move parallel to the direction of wave travel (e.g., sound waves in air).

• Mechanical Waves: Require a material medium (e.g., string, air, water). The speed of a mechanical wave depends on the properties of the medium, not the wave itself.

• Electromagnetic Waves: Do not require a medium and can travel through a vacuum. All electromagnetic waves travel at the same speed in a vacuum: $c = 3.00 \times 10^8$ m/s.

Key Equations:

• Speed of a wave on a string: $v_{string} = \sqrt{\frac{T}{\mu}}$

• Speed of sound in a gas: $v_{sound} = \sqrt{\frac{\gamma RT}{M}}$

Example: Sound waves in air are longitudinal mechanical waves, while light is an electromagnetic wave.

Recap: Light and the Electromagnetic Spectrum

The visible spectrum is a small part of the much broader electromagnetic spectrum, which includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Each type of wave is characterized by its frequency and wavelength.

Recap: Mathematical Representation of Waves

Sinusoidal waves are produced by sources undergoing simple harmonic motion. The general equation for a sinusoidal wave is:

• $y(x, t) = A \cos\left[2\pi \left(\frac{x}{\lambda} - \frac{t}{T}\right)\right]$

• Where $A$ is amplitude, $\lambda$ is wavelength, $T$ is period, $x$ is position, and $t$ is time.

• Wave speed: $v = \lambda f$

Section 15.5: Energy and Intensity

Energy and Intensity of Waves

Waves transfer energy from one location to another. The power of a wave is the rate at which energy is transferred, measured in watts (W). Intensity is the power per unit area and describes how concentrated the wave's energy is.

• Intensity formula: $I = \frac{P}{a}$, where $P$ is power and $a$ is area.

• SI units: W/m2

• A wave focused on a small area has higher intensity than if spread out over a larger area.

Example: A laser pointer emitting 1.0 mW of light into a 1.0 mm diameter beam has an intensity:

• $I = \frac{P}{\pi r^2} = \frac{0.0010\,\text{W}}{\pi (0.00050\,\text{m})^2} = 1300\,\text{W/m}^2$

This is comparable to sunlight at noon, showing how a small area can yield high intensity.

Circular, Spherical, and Plane Waves

• Circular Waves: Two-dimensional waves spreading across a surface (e.g., ripples in a pond). Crests are wave fronts, spaced one wavelength apart.

• Spherical Waves: Three-dimensional waves (e.g., sound, light) with crests forming spherical shells.

• Plane Waves: At large distances from the source, small segments of spherical wave fronts appear flat and are called plane waves.

Intensity of Spherical Waves

• For a source radiating uniformly in all directions, the surface area is $4\pi r^2$.

• Intensity at distance $r$: $I = \frac{P_{source}}{4\pi r^2}$

• Intensity ratio at two points: $\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$

Example: Solar cells in space: If sunlight intensity is 1300 W/m2 at Earth's distance, at Neptune (30 times farther), intensity drops by a factor of $30^2 = 900$.

Section 15.6: Loudness of Sound

Loudness and the Decibel Scale

Loudness is the human perception of sound intensity. The sound intensity level is measured in decibels (dB) using a logarithmic scale, reflecting the nonlinear response of human hearing.

• Increasing sound intensity by a factor of 10 increases perceived loudness by a factor of about 2.

• Threshold of hearing: $I_0 = 1.0 \times 10^{-12}$ W/m2

• Sound intensity level: $\beta = (10\,\text{dB}) \log_{10}\left(\frac{I}{I_0}\right)$

• At the threshold of hearing ($I = I_0$), $\beta = 0$ dB.

• To find intensity from dB: $I = I_0 \times 10^{\beta/10}$

Table: Sound Intensity Levels of Common Sounds

Sound	B (dB)	I (W/m2)
Threshold of hearing	0	1.0×10-12
Person breathing, at 3 m	10	1.0×10-11
A whisper, at 1 m	20	1.0×10-10
Classroom during test, no talking	30	1.0×10-9
Residential street, no traffic	40	1.0×10-8
Quiet restaurant	50	1.0×10-7
Normal conversation, at 1 m	60	1.0×10-6
Busy traffic	70	1.0×10-5
Vacuum cleaner, for user	80	1.0×10-4
Niagara Falls, at viewpoint	90	1.0×10-3
Pneumatic hammer, at 2 m	100	0.010
Home stereo at max volume	110	0.10
Rock concert	120	1.0
Threshold of pain	130	10

Example: A person shouting with 1.0 W of power at 1.0 m distance:

• $I = \frac{1.0\,\text{W}}{4\pi (1.0\,\text{m})^2} = 0.080\,\text{W/m}^2$

• $\beta = (10\,\text{dB}) \log_{10}\left(\frac{0.080}{1.0\times10^{-12}}\right) = 110\,\text{dB}$

This is extremely loud and can cause hearing damage with prolonged exposure.

Section 15.7: The Doppler Effect and Shock Waves

The Doppler Effect

The Doppler effect is the change in observed frequency of a wave due to the relative motion of the source and observer. It explains phenomena such as the pitch change of a passing siren.

• Approaching source: Observed frequency increases.

• Receding source: Observed frequency decreases.

Key Equations:

• For a moving source and stationary observer:

• Approaching: $f' = \frac{f}{1 - v_s/v}$

• Receding: $f' = \frac{f}{1 + v_s/v}$

• Where $f$ is the emitted frequency, $v_s$ is the speed of the source, and $v$ is the speed of sound.

Example: A police siren emits 550 Hz approaching and 450 Hz receding. The speed of the police car can be found using the Doppler equations.

The Doppler Effect for Light Waves

• If a light source is moving away, the observed wavelength increases (red shift).

• If moving toward, the wavelength decreases (blue shift).

• All distant galaxies are red shifted, supporting the Big Bang theory.

Frequency Shift on Reflection (Double Doppler Shift)

• When a wave reflects off a moving object, the frequency shift is doubled.

• Used in Doppler ultrasound to measure blood flow.

• Frequency shift: $\Delta f = 2f \frac{v_{object}}{v_{wave}}$

Shock Waves

• A shock wave forms when a source moves faster than the wave speed in the medium (supersonic).

• The overlapping waves create a large amplitude wave, perceived as a sonic boom.

• Other examples include the wake of a boat.

Summary of Key Concepts

• Intensity: $I = \frac{P}{a}$

• Spherical wave intensity: $I = \frac{P_{source}}{4\pi r^2}$

• Sound intensity level (dB): $\beta = (10\,\text{dB}) \log_{10}\left(\frac{I}{I_0}\right)$

• Doppler effect: Frequency shifts due to relative motion of source and observer.

• Shock waves: Produced when source speed exceeds wave speed.

Applications: These principles are essential for understanding sound, light, medical imaging (ultrasound), astronomy (red shift), and engineering (sonic booms).