understand the use of the time-base and y-gain of a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

Progressive Waves – Using a Cathode‑Ray Oscilloscope (CRO)

1. Aim

To use the horizontal time‑base and vertical y‑gain controls of a CRO to obtain the frequency and amplitude of a travelling (progressive) wave, and to relate these measurements to the quantities required by the Cambridge 9702 – Waves syllabus (7.1‑7.3).

2. Key Wave Quantities (Syllabus 7.1)

3. CRO Controls Relevant to Wave Measurements

4. Determining Frequency from the Time‑Base

1. Display a sinusoidal voltage on the screen.
2. Count the number of horizontal divisions that contain one complete cycle – call this L_h (div).
3. Calculate the period:
   

   \$T = L_{h}\,S\$

4. Frequency follows directly:
   

   \$f = \frac{1}{T}= \frac{1}{L_{h}\,S}\$

5. Determining Amplitude from the Y‑Gain

1. Measure the vertical peak‑to‑peak height in divisions – L_v (div).
2. Convert to peak‑to‑peak voltage:
   

   \$V{pp}=L{v}\,V_{div}\$

3. For a sinusoid the amplitude (peak value) is half the peak‑to‑peak voltage:
   

   \$A = \frac{V{pp}}{2}= \frac{L{v}\,V_{div}}{2}\$

6. From Frequency and Wavelength to Wave Speed (v = f λ)

The CRO gives the frequency of the electrical signal that is produced by the mechanical wave source. The wavelength must be obtained from a physical measurement on the medium.

6.1 Measuring the Wavelength

• Node‑spacing method (standing wave on a string): distance between two adjacent nodes = λ/2 →
  

  \$\lambda = 2\,(\text{node spacing})\$

• Mode‑number method: if the string length is L and the standing‑wave mode number is n (number of half‑wavelengths), then
  

  \$\lambda = \frac{2L}{n}\$

6.2 Derivation of v = f λ using CRO data

From the CRO we have measured f. The wavelength is obtained as described above. Substituting the two measured quantities gives the wave speed:

\$v = f\,\lambda = \frac{1}{L_{h}S}\;\times\;\lambda\$

This explicit link – “frequency from the time‑base, wavelength from the medium → wave speed” – satisfies the syllabus requirement that students *show* how the CRO data lead to the wave‑speed equation.

7. Intensity and Its Dependence on Amplitude

For a transverse wave on a string the intensity (average power per unit cross‑sectional area) is defined by the syllabus as

\$I = \frac{P}{A_{\text{cross}}}\$

where P is the average power transmitted and A_cross the cross‑sectional area of the string. Using the standard result for a sinusoidal travelling wave on a stretched string, the average power is

\$P = \tfrac{1}{2}\,\rho\,v\,A^{2}\$

Dividing by the (constant) cross‑sectional area gives the intensity formula used in the notes:

\$I = \frac{1}{2}\,\rho\,v\,A^{2}\qquad\text{(∝ A²)}\$

Thus, any change in the measured vertical division count L_v (and therefore in A) leads to a quadratic change in intensity – a point that often appears in AO2 exam questions.

8. Uncertainty Propagation (AO2)

When reporting results, include the uncertainties arising from reading divisions on the CRO and from the physical measurement of wavelength.

• Horizontal division count: ΔL_h ≈ ±0.1 div
• Vertical division count: ΔL_v ≈ ±0.1 div
• Time‑base setting: ΔS (usually given by the instrument, e.g. ±1 % of the selected value)
• Y‑gain setting: ΔV_div (similarly ±1 %)
• Wavelength (node spacing or mode‑number method): Δλ from ruler accuracy (±0.5 mm) or from the uncertainty in n.

Using standard propagation rules:

\[

\frac{\Delta f}{f}= \sqrt{\left(\frac{\Delta Lh}{Lh}\right)^2+\left(\frac{\Delta S}{S}\right)^2},

\qquad

\frac{\Delta A}{A}= \sqrt{\left(\frac{\Delta Lv}{Lv}\right)^2+\left(\frac{\Delta V{div}}{V{div}}\right)^2},

\]

\[

\frac{\Delta v}{v}= \sqrt{\left(\frac{\Delta f}{f}\right)^2+\left(\frac{\Delta \lambda}{\lambda}\right)^2},

\qquad

\frac{\Delta I}{I}= \sqrt{\left(\frac{\Delta \rho}{\rho}\right)^2+\left(\frac{\Delta v}{v}\right)^2+4\left(\frac{\Delta A}{A}\right)^2}.

\]

9. Transverse vs. Longitudinal Waves (Syllabus 7.2)

10. Doppler Effect (Syllabus 7.3 – optional extension)

If the source moves with speed v_s relative to the medium, the observed frequency on the CRO changes to

\$f{o}=f{s}\,\frac{v}{v\pm v_{s}}\$

• Use “+” when the source moves away from the observer, “–” when it moves towards the observer.
• The CRO directly displays the shifted frequency f_o; rearranging the formula yields the original source frequency f_s.

11. Worked Example (All Quantities)

Given

• Time‑base S = 0.5 ms/div
• Y‑gain V_div = 2 V/div
• Horizontal length of one cycle L_h = 4 div
• Vertical peak‑to‑peak height L_v = 3 div
• String length L = 0.80 m; node spacing = 0.10 m → λ = 0.20 m
• Linear mass density ρ = 0.02 kg m⁻¹
• Transducer conversion: 1 V ↔ 0.5 mm displacement

1. Period:  T = 4 div × 0.5 ms/div = 2 ms
2. Frequency:  f = 1/T = 500 Hz
3. Peak‑to‑peak voltage:  V_pp = 3 div × 2 V/div = 6 V
4. Amplitude (electrical):  A_e = 6 V/2 = 3 V
   

   Mechanical amplitude:  A = 3 V × 0.5 mm V⁻¹ = 1.5 mm = 1.5 × 10⁻³ m

5. Wavelength:  λ = 2 × 0.10 m = 0.20 m
6. Wave speed:  v = f λ = 500 Hz × 0.20 m = 100 m s⁻¹
7. Intensity:  I = ½ ρ v A² = ½ (0.02)(100)(1.5 × 10⁻³)² ≈ 2.3 × 10⁻⁶ W m⁻¹

12. Practical Tips (AO1 & AO2)

• Calibrate the CRO with the built‑in calibration signal before taking data.
• Adjust the time‑base so that one full cycle occupies 2–4 horizontal divisions – this minimises reading error.
• Set the y‑gain so the waveform uses about 2–4 vertical divisions peak‑to‑peak without clipping.
• Use the CRO’s cursors or the “measure” function for division counts; typical reading uncertainty ±0.1 div.
• When measuring wavelength on a string, keep the tension constant and use a ruler with millimetre accuracy.
• Record the uncertainties of S, V_div, L_h, L_v and λ; propagate them as shown in Section 8.

13. Summary Table