Fundamental Wave Properties to Know for AP Physics 2 (2025)

1. What You Need to Know

Waves show up everywhere in AP Physics 2 (sound, light, interference/diffraction, Doppler, standing waves). The exam mostly tests whether you can connect what a wave “looks like” to what it does using a small set of core definitions and relationships.

The core idea

A wave is a traveling disturbance that transfers energy and momentum without (necessarily) transferring matter over long distances.

The single most-used relationship

Wave speed ties together wavelength and frequency:

v = f\lambda

• Speed v: how fast the pattern moves (set by the medium, not by how hard you shake it).
• Frequency f: oscillations per second (set by the source).
• Wavelength \lambda: distance between repeating points.

Critical reminder: When a wave enters a new medium, f stays the same (source-controlled), while v and \lambda can change.

What “fundamental wave properties” means on AP Physics 2

You should be able to:

• Translate between graphs and parameters: amplitude, wavelength, period, phase.
• Use the math form of a sinusoidal wave: amplitude, wavenumber, angular frequency, phase.
• Apply superposition to get interference, beats, standing waves.
• Use intensity/power ideas (especially for sound and light).
• Handle boundary behavior (reflection phase inversion) and medium changes (speed and wavelength changes).

2. Step-by-Step Breakdown

A) Any basic traveling-wave calculation (fast method)

1. Identify what’s given: f or T, \lambda, v, or a graph.

2. Convert if needed: f = \frac{1}{T}.

3. Use the anchor equation: v = f\lambda.

4. If they give a sinusoidal equation, match it to:

   y(x,t) = A\sin(kx \mp \omega t + \phi)

   and extract:

   • amplitude A
   • wavenumber k = \frac{2\pi}{\lambda}
   • angular frequency \omega = 2\pi f

5. Check units and physical sense (e.g., higher f at fixed v means smaller \lambda).

Mini-check example (graph-free):

• If f = 50\ \text{Hz} and \lambda = 2.0\ \text{m}, then

  v = f\lambda = (50)(2.0) = 100\ \text{m/s}

B) Interference decision tree (two-source problems)

1. Decide if sources are coherent (constant phase difference). If yes, stable interference.

2. Compute path difference \Delta r = r_2 - r_1.

3. Determine interference condition (assuming sources are in phase):

   • Constructive: \Delta r = m\lambda
   • Destructive: \Delta r = \left(m + \tfrac{1}{2}\right)\lambda

4. If they ask about phase, use:

   \Delta \phi = \frac{2\pi}{\lambda}\Delta r + \Delta \phi_0

5. If they ask about resulting intensity (more advanced but testable):

   I_{\text{tot}} = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos(\Delta\phi)

Decision point: If there’s a reflection off a boundary, include possible phase inversion (see Section 3).

C) Standing waves on strings/pipes (quick method)

1. Identify boundary type:

   • String fixed-fixed or open-open pipe: nodes at both ends.
   • open-closed pipe: node at closed end, antinode at open end.

2. Write allowed wavelengths:

   • Fixed-fixed or open-open:

     L = \frac{n\lambda}{2} \Rightarrow \lambda_n = \frac{2L}{n}

   • Open-closed:

     L = \frac{(2n-1)\lambda}{4} \Rightarrow \lambda_n = \frac{4L}{2n-1}

3. Convert to frequencies using f_n = \frac{v}{\lambda_n}.

Mini-check example (string):

• If L = 0.80\ \text{m}, v = 120\ \text{m/s}, then fundamental n=1:

  f_1 = \frac{v}{2L} = \frac{120}{1.6} = 75\ \text{Hz}

3. Key Formulas, Rules & Facts

A) Core definitions & kinematics of waves

Phase speed vs particle speed (don’t mix):

• v in v=f\lambda is wave pattern speed.
• Particles of the medium oscillate with max speed v_{\text{particle,max}} = \omega A (for sinusoidal motion).

B) Wave speed depends on medium

C) Intensity, power, and amplitude (high yield)

Decibel quick facts:

• +10 dB means I is multiplied by 10.
• +20 dB means I is multiplied by 100.

D) Superposition, interference, beats

If two waves of equal amplitude A are in phase, resultant amplitude is 2A.

E) Standing waves (strings and air columns)

F) Reflection at boundaries (phase inversion)

• Fixed end reflection: displacement flips sign → \pi phase shift.
• Free end reflection: no inversion → no phase shift.

This matters when you’re deciding constructive vs destructive interference after reflection.

G) Refraction & wavelength change (wave property, not just “optics”)

• Frequency stays constant across boundary:

  f_1 = f_2

• So if speed changes, wavelength changes:

  v_1 = f\lambda_1,\quad v_2 = f\lambda_2 \Rightarrow \frac{\lambda_2}{\lambda_1} = \frac{v_2}{v_1}

• For light in a medium:

  n = \frac{c}{v},\quad \lambda_{\text{medium}} = \frac{\lambda_0}{n}

H) Doppler effect (sound is most common)

For sound in air with wave speed v:

f' = f\left(\frac{v \pm v_o}{v \mp v_s}\right)

• v_o: observer speed (use + when observer moves toward source)
• v_s: source speed (use − in denominator when source moves toward observer)

4. Examples & Applications

Example 1: Extracting wave info from a sinusoidal equation

Given:

y(x,t) = 0.020\sin\left(4\pi x - 200\pi t\right)

Setup & key insights:

• Amplitude A = 0.020\ \text{m}.

• Wavenumber k = 4\pi\ \text{rad/m} \Rightarrow \lambda = \frac{2\pi}{k} = \frac{2\pi}{4\pi} = 0.50\ \text{m}.

• Angular frequency \omega = 200\pi\ \text{rad/s} \Rightarrow f = \frac{\omega}{2\pi} = \frac{200\pi}{2\pi} = 100\ \text{Hz}.

• Wave speed:

  v = f\lambda = (100)(0.50) = 50\ \text{m/s}

Direction: kx-\omega t means it travels in +x.

Example 2: Interference from two in-phase sources

Two speakers emit in phase at frequency f = 680\ \text{Hz}. Speed of sound v = 340\ \text{m/s}.

• Wavelength:

  \lambda = \frac{v}{f} = \frac{340}{680} = 0.50\ \text{m}

At a point where path difference is \Delta r = 0.75\ \text{m}:

• Compare with wavelength:

  \Delta r = 0.75\ \text{m} = 1.5\lambda = \left(1 + \tfrac{1}{2}\right)\lambda

So it’s destructive interference.

Example 3: Standing wave harmonic on an open-closed pipe

An open-closed tube has length L = 0.30\ \text{m} with sound speed v = 340\ \text{m/s}.

Allowed frequencies:

f_n = \frac{(2n-1)v}{4L}

• Fundamental (first harmonic, n=1):

  f_1 = \frac{(1)(340)}{4(0.30)} \approx 283\ \text{Hz}

• Next allowed (third harmonic, n=2):

  f_2 = \frac{(3)(340)}{4(0.30)} \approx 850\ \text{Hz}

Key insight: open-closed supports only odd harmonics.

Example 4: Doppler effect with moving source

A siren emits f = 1000\ \text{Hz}. The source moves toward a stationary observer at v_s = 20\ \text{m/s}. Take v = 340\ \text{m/s}.

Use Doppler (observer stationary, so v_o=0):

f' = f\left(\frac{v}{v - v_s}\right) = 1000\left(\frac{340}{340-20}\right) = 1000\left(\frac{340}{320}\right) \approx 1063\ \text{Hz}

Key insight: moving source changes the wavelength in front of it, raising the observed frequency.

5. Common Mistakes & Traps

1. Mixing up what changes at a boundary

   • Wrong: saying light’s f changes when entering glass.
   • Why wrong: frequency is set by the source; boundary conditions keep oscillation rate continuous.
   • Fix: use f_1=f_2 and adjust v and \lambda.

2. Confusing angular frequency with frequency

   • Wrong: treating \omega as Hz.
   • Why wrong: \omega is rad/s.
   • Fix: always use \omega = 2\pi f.

3. Confusing wavenumber with wavelength

   • Wrong: using k = \lambda.
   • Why wrong: k is spatial angular frequency.
   • Fix: k = \frac{2\pi}{\lambda}.

4. Adding intensities instead of displacements (or vice versa)

   • Wrong: claiming total displacement is I_1 + I_2.
   • Why wrong: superposition adds displacements: y_{\text{tot}} = y_1 + y_2.
   • Fix: find displacement/phase first; only convert to intensity if asked.

5. Forgetting phase inversion on reflection

   • Wrong: treating reflection from a fixed end as if it returns in phase.
   • Why wrong: fixed boundary forces displacement to be zero → inversion.
   • Fix: remember: fixed end = \pi shift; free end = none.

6. Misidentifying harmonics for open-closed pipes

   • Wrong: using f_n = \frac{nv}{2L} for an open-closed tube.
   • Why wrong: boundary conditions are different.
   • Fix: open-closed uses f_n = \frac{(2n-1)v}{4L}.

7. Doppler sign errors

   • Wrong: memorizing a formula but flipping signs randomly.
   • Why wrong: the observed frequency increases only when the distance between wavefronts reaching you decreases.
   • Fix: use the “toward increases” logic: observer toward ⇒ numerator bigger; source toward ⇒ denominator smaller.

8. Decibel misconceptions

   • Wrong: thinking +10 dB means “10× louder” (as a perception claim).
   • Why wrong: dB is a logarithmic measure of intensity ratio; perceived loudness is not exactly intensity.
   • Fix: interpret strictly: +10 dB ⇒ I is 10×.

6. Memory Aids & Quick Tricks

7. Quick Review Checklist

• You can use v = f\lambda instantly and correctly.
• You remember: boundary change ⇒ f constant, v and \lambda can change.
• You can extract A, k, \omega, \lambda, f, and direction from y(x,t)=A\sin(kx\mp\omega t+\phi).
• You know interference conditions: constructive \Delta r=m\lambda, destructive \Delta r=\left(m+\tfrac{1}{2}\right)\lambda (for in-phase sources).
• You apply reflection phase shifts: fixed end inverts, free end doesn’t.
• You can write standing wave frequencies for:
  • fixed-fixed/open-open: f_n=\frac{nv}{2L}
  • open-closed: f_n=\frac{(2n-1)v}{4L}
• You can use intensity rules: I=\frac{P}{A}, spherical spreading I\propto\frac{1}{r^2}, and \beta=10\log_{10}(I/I_0).
• You can handle Doppler with the “toward increases” sign logic.

You’ve only got a handful of wave tools—use them confidently and you’ll catch most AP wave questions quickly.