AP Physics 2- 14.1 Properties of Wave Pulses and Waves- Study Notes- New Syllabus
AP Physics 2- 14.1 Properties of Wave Pulses and Waves – Study Notes
AP Physics 2- 14.1 Properties of Wave Pulses and Waves – Study Notes – per latest Syllabus.
Key Concepts:
- Physical Properties of Waves
- Physical Properties of Waves Pulses
Physical Properties of Waves
A wave is a disturbance that transfers energy without the transport of matter. Waves are characterized by measurable physical properties that describe their behavior.
Wave Properties:
Wavelength (\(\mathrm{\lambda}\)):
The distance between two consecutive crests, troughs, or compressions. Units: \(\mathrm{m}\).
Frequency (\(\mathrm{f}\)):
Number of oscillations per second.
\(\mathrm{f = \dfrac{1}{T}}\), where \(\mathrm{T}\) = time period. Units: \(\mathrm{Hz}\).
Wave Speed (\(\mathrm{v}\)):
The rate at which the wave propagates:
\(\mathrm{v = f \lambda}\).
Amplitude (\(\mathrm{A}\)):
Maximum displacement of the medium from equilibrium. Determines the energy of the wave.
Phase:
Describes the position of a point within the wave cycle (measured in radians or degrees).
Waveform:
The shape of the wave (e.g., sinusoidal, square, triangular).
Transverse vs Longitudinal Waves:
Transverse Waves:
The oscillations of the medium are perpendicular to the direction of wave propagation.
- Examples: light waves, water surface waves, waves on a string.
- They are characterized by crests (maximum upward displacement) and troughs (maximum downward displacement).
Mathematically described as:
$ \mathrm{y(x,t) = A \sin(kx – \omega t + \phi)} $
where \(\mathrm{A}\) is amplitude, \(\mathrm{k}\) is wave number, \(\mathrm{\omega}\) is angular frequency.
Longitudinal Waves:
The oscillations of the medium are parallel to the direction of wave propagation.
- Examples: sound waves in air, compression waves in springs.
- They consist of compressions (high-density regions) and rarefactions (low-density regions).
- Wave motion transfers energy through alternating pressure variations.
Comparison:
| Feature | Transverse Waves | Longitudinal Waves |
|---|---|---|
| Oscillation Direction | Perpendicular to wave motion | Parallel to wave motion |
| Wave Parts | Crests & troughs | Compressions & rarefactions |
| Examples | Light, water waves, string vibrations | Sound in air, slinky compression |
| Medium Requirement | Can travel through solids, some can through vacuum (e.g., EM waves) | Require a medium (cannot travel in vacuum) |
Energy in a Wave:
The energy carried by a wave is proportional to the square of its amplitude:
$ \mathrm{E \propto A^2} $
Wave Speed on a String:
The speed at which a wave pulse propagates along a string depends on the tension in the string (\(\mathrm{F_T}\)) and the mass per unit length of the string (\(\mathrm{\mu = \dfrac{m}{L}}\)):
$ \mathrm{v = \sqrt{\dfrac{F_T}{\mu}}} $
Increasing tension \(\mathrm{F_T}\) increases wave speed.
Increasing mass per length \(\mathrm{\mu}\) decreases wave speed.
Example :
A wave has a frequency of \(\mathrm{500 \, Hz}\) and wavelength \(\mathrm{0.68 \, m}\). Find its speed.
▶️ Answer/Explanation
Step 1: Formula: \(\mathrm{v = f \lambda}\)
Step 2: Substitute: \(\mathrm{v = (500)(0.68) = 340 \, m/s}\)
Final Answer: Wave speed = \(\mathrm{340 \, m/s}\).
Example :
A string of length \(\mathrm{2.0 \, m}\) has a mass of \(\mathrm{0.010 \, kg}\) and is under a tension of \(\mathrm{50 \, N}\). Find the speed of a wave pulse on the string.
▶️ Answer/Explanation
Step 1: Mass per unit length: \(\mathrm{\mu = \dfrac{m}{L} = \dfrac{0.010}{2.0} = 0.005 \, kg/m}\)
Step 2: Wave speed: \(\mathrm{v = \sqrt{\dfrac{F_T}{\mu}} = \sqrt{\dfrac{50}{0.005}} = \sqrt{10000} = 100 \, m/s}\)
Final Answer: Wave speed = \(\mathrm{100 \, m/s}\).
Physical Properties of Wave Pulses
A wave pulse is a single disturbance that travels through a medium, transferring energy from one point to another without transporting matter.
Properties of a Wave Pulse:
Amplitude (\(\mathrm{A}\)):
The maximum displacement of the medium from its equilibrium position. Determines the energy carried by the pulse:
$ \mathrm{E \propto A^2} $
Pulse Length:
The spatial extent of the disturbance (distance over which the medium is displaced).
Pulse Speed (\(\mathrm{v}\)):
The rate at which the pulse propagates through the medium. For a string:
$ \mathrm{v = \sqrt{\dfrac{F_T}{\mu}}} $
where
\(\mathrm{F_T}\) = tension in the string and
\(\mathrm{\mu = \dfrac{m}{L}}\) = mass per unit length.
Direction of Motion:
The direction in which the disturbance travels (left/right along the medium).
Reflection and Inversion:
- At a fixed end, the reflected pulse inverts (flips upside down).
- At a free end, the reflected pulse remains upright.
Superposition Principle:
When two or more pulses meet, the resulting displacement is the algebraic sum of the individual displacements (constructive or destructive interference).
Example :
A string of length \(\mathrm{3.0 \, m}\) has a mass of \(\mathrm{0.015 \, kg}\). It is held under a tension of \(\mathrm{45 \, N}\). Find the speed of a wave pulse on the string.
▶️ Answer/Explanation
Step 1: Mass per unit length: \(\mathrm{\mu = \dfrac{m}{L} = \dfrac{0.015}{3.0} = 0.005 \, kg/m}\)
Step 2: Wave speed: \(\mathrm{v = \sqrt{\dfrac{F_T}{\mu}} = \sqrt{\dfrac{45}{0.005}} = \sqrt{9000} \approx 95 \, m/s}\)
Final Answer: Wave pulse speed = \(\mathrm{95 \, m/s}\).
Example :
A pulse with amplitude \(\mathrm{5.0 \, cm}\) carries an energy \(\mathrm{E}\). If the amplitude is doubled, what happens to the energy of the pulse?
▶️ Answer/Explanation
Step 1: Relation: \(\mathrm{E \propto A^2}\)
Step 2: Compare energies: \(\mathrm{\dfrac{E_{new}}{E_{old}} = \left(\dfrac{A_{new}}{A_{old}}\right)^2 = \left(\dfrac{2A}{A}\right)^2 = 4}\)
Final Answer: The energy becomes \(\mathrm{4E}\), i.e., quadrupled.