Notes for Hw 3

An object vibrates, causing a disturbance in the surrounding air creating sound.
Movement in air particles occurs, generating sound waves associated with the vibrations.
As a result, sound waves begin to propagate through the medium.

Pitch refers to how high or low a sound is.
- Determined by the frequency of the sound waves; higher frequency corresponds to a higher pitch.
Volume signifies how loud or soft a sound is.
- Influenced by the amplitude of the sound waves; higher amplitude yields a louder sound.
Low frequency notes: Correspond to slower vibrations.
High frequency notes: Correspond to faster vibrations.
- Example: Higher frequency ($f$) equals higher pitch, higher amplitude ($A$) equals louder sound.

Rarefactions and Compressions:
- Rarefactions are regions where particles are spread apart.
- Compressions are regions where particles are close together.
Direction of Sound Wave: Illustrates the path that sound takes as it propagates.

Crest: The highest point of the wave.
Trough: The lowest point of the wave.
Amplitude: The height of the wave from the equilibrium position to the crest (or trough), indicating the energy of the wave.
Wavelength ($ ext{λ}$): Distance between consecutive crests (or troughs).

V = f imes ext{λ} where:
- V = wave speed (m/s),
- f = frequency (Hz),
- ext{λ} = wavelength (m).

Temperature of the medium: Sound travels faster in warm air than in cold air; specific speeds are approximated as follows:
- Sound Speed in Air (0°C): 331 m/s
- Sound Speed in Air (20°C): 344 m/s
Medium Density: Greater density slows sound down (kg/m³).
Bulk Modulus: Indicates how resistant a material is to compression; higher bulk modulus equates to faster sound speed.
Example speed values for various materials (speed in m/s):
- Air (0°C): 331
- Air (20°C): 344
- Helium (20°C): 999
- Hydrogen (20°C): 1330
- Water (0°C, liquid): 1400
- Water (20°C): 1480
- Aluminium: 6420
- Steel: 5940
- Lead: 1960

Definition: Perception of the sound volume by the ear.
- Measured in decibels (dB) or bels (B).
- Depends on the sensitivity of the ear, varying from person to person.

Definition: The power of the wave transmitted per unit area.
- Measured in units of watts per square meter (W/m²).
- Intensity increases with the increasing amplitude of the sound wave.
Relationship:
Intensity ($I$) is given by:
I = rac{P}{A}
where:
- P = Power (in watts), A = Area (in square meters).

Loudness is expressed on a logarithmic scale while intensity is typically represented on a linear scale.
- Consequentially, doubling the intensity does not equate to it feeling twice as loud to the ear.

Specific sound levels and associated risks:
- 150 dB: Jet Plane, Explosion – High Risk, Hearing damage.
- 130 dB: Formula 1 – High Noise Level, mandatory protection.
- 110 dB: Airplane Taking Off, Jackhammer – Protection advised.
- 100 dB: Loud environments like concert or ambulance sirens – Warning for hearing damage.
- 85 dB: Drilling or sanding; necessary precautions advised.
- 70 dB: Office sounds, safe listening level.
- 60 dB: Conversations; considered normal levels.
Human hearing range: 20 Hz to 20,000 Hz.
- Frequency increases (+f) correlates with pitch.

Sound waves are energy transporters without the transportation of mass.
- Intensity is a measure of the power transmitted by a wave per area.
Intensity Formula:
I = rac{P}{A}
where:
- I = Intensity (W/m²), P = Power, A = Area.
Sound intensity decreases inversely with the square of distance from the source:
I(r) = rac{P}{4πr^2}
Area of dispersion: Energy spreads uniformly in all directions from the sound source.

The modes of vibration in standing waves are based on the harmonic number denoted as: f_n = n rac{v}{2L} where:
- n = harmonic number
- L = length of the vibrating string

Example harmonic numbers (1 through 5 for a vibrating string):
- 1st harmonic (fundamental): Frequency f_1 = rac{v}{2L}
- 2nd harmonic (1st overtone): Frequency f_2 = rac{v}{L}
Nodes and Antinodes: The number of nodes present determines the mode of vibration
- ext{Number of nodes} = n + 1

Vibrations of strings create standing waves which determine the sound produced:
- Fundamental frequency depends upon the tension, length, and linear mass density of the string.
Tension: Changes the pitch while length and mass typically remain constant.
Timbre is influenced by the harmonics created which result from how the string is excited (plucked, bowed, struck etc.).
Various string instruments include:
- Violin, Guitar, Cello, Ukrainian Bandur, etc.

Wave Speed calculation depends on tension and mass:
V = rac{F_T}{ ext{m}}
- where:
- F_T = tension force (N),
- m = mass (kg).
The governing principles for standing waves ensure that various harmonics produce different sounds depending on instrument design and mechanics.
Please note all content covers essential aspects related to sound energy and how it translates into hearing phenomena, properties of sound waves, and their applications in real-world contexts like music and environmental science.