Sound Waves, Intensity level, Decibels, Beat Frequency, Doppler Effect, Open Organ Pipe - Physics

A sound wave is a longitudinal wave, where oscillations are parallel to the wave direction, unlike transverse waves. It's essentially a pressure wave, creating regions of high and low pressure (compressions and rarefactions) as it travels through a medium. Sound requires a medium to travel and cannot propagate through a vacuum.

The speed of sound in air is approximately 343 m/s at 20°C. This speed is affected by factors like temperature, density, and the material through which it travels. For solids, speed is related to elastic modulus and density, while for fluids, it involves the bulk modulus. Generally, sound travels faster in solids than in liquids, and faster in liquids than in gases. Higher temperatures increase the speed of sound due to faster molecular movement.

Subsonic refers to speeds less than the speed of sound, while supersonic refers to speeds greater than it, often creating a sonic boom. The human audible range for sound is from 20 Hz to 20,000 Hz (20 kHz), though this upper limit decreases with age. Frequencies below 20 Hz are infrasonic, and above 20 kHz are ultrasonic, used in applications like sonar.

A sound wave can be represented as a sinusoidal function, with amplitude as its height and wavelength as the distance between two crests. The period is the time for one complete cycle, and frequency is the number of cycles per second (reciprocal of the period). The speed of a wave (V) is the product of its wavelength (λ) and frequency (f): V = λf.

Several problems are solved, demonstrating how to calculate frequency given wavelength and speed, and how to determine the time it takes for sound to travel a certain distance. An example calculates the range of a lake using the time taken for an echo to return, and the speed of sound at a given temperature.

Intensity is defined as power divided by area, typically the surface area of a sphere (4πr²). It's measured in watts per square meter. The inverse square law states that intensity is inversely proportional to the square of the distance from the source. Doubling the distance reduces intensity to one-fourth. Power represents the rate of energy transfer.

The sound intensity level is measured in decibels (dB), calculated using the formula β = 10 log(I/I₀), where I₀ is the threshold of hearing (1 × 10⁻¹² W/m²). A 10-fold increase in intensity results in a 10 dB increase in intensity level, making the sound appear twice as loud. Various problems illustrate calculating intensity from intensity level and vice-versa, and how intensity changes with distance.

Intensity is directly proportional to the square of both frequency and amplitude. This means if frequency or amplitude is doubled, intensity increases by a factor of four. The kinetic energy of oscillating air molecules due to sound waves is linked to this relationship. This section includes calculations for displacement amplitude and the impact of changes in frequency and amplitude on intensity.

Standing waves are formed when a string is fixed at both ends. The velocity of a wave on a string depends on tension and linear density. Different patterns (harmonics) can be produced, with the fundamental frequency being the first harmonic. Overtones are higher harmonics. Formulas are provided to relate wavelength, frequency, length of the string, and the harmonic number.

The behavior of sound waves in organ pipes depends on whether they are open or closed at the ends. For open pipes, both ends are anti-nodes for displacement, and the harmonics follow the same rules as a string fixed at both ends (n can be any integer). For closed pipes (one end closed, one open), the closed end is a node and the open end is an anti-node. Only odd harmonics are present (n = 1, 3, 5,...).

The principle of superposition states that waves add up. Constructive interference occurs when waves are in phase, resulting in a larger amplitude. Destructive interference occurs when waves are out of phase, leading to smaller or canceled amplitudes. For destructive interference, path difference is an odd multiple of half a wavelength, while for constructive interference, it's an integer multiple of a wavelength. Examples involve calculating distances for these phenomena.

When two sound sources produce slightly different frequencies, a beat frequency is heard. This is the absolute difference between the two frequencies. The beat frequency represents the number of times per second constructive interference occurs, resulting in periodic loudness variations. Problems involve determining possible frequencies given a beat frequency.

The Doppler effect describes the change in detected frequency when there is relative motion between a sound source and an observer. If the source moves towards the observer, the frequency increases (higher pitch); if it moves away, the frequency decreases (lower pitch). This is due to the compression or stretching of sound waves. A detailed formula is provided to calculate the observed frequency, accounting for the speeds of both the source and the observer.

Several problems apply the Doppler effect formula to various scenarios, including an ambulance siren, a moving jeep, and a police radar gun. The radar gun problem involves a two-part Doppler shift: sound traveling from the gun to the vehicle, and then the reflected sound traveling back to the gun, effectively changing the source and observer in the second part of the calculation.