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.