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. Objective

To understand how the time‑base (horizontal) and y‑gain (vertical) controls of a CRO can be used to determine the frequency and amplitude of a travelling (progressive) wave.

2. Key Concepts

• A progressive wave can be represented by a sinusoidal function: \$y(x,t)=A\sin(kx-\omega t+\phi)\$ where \$A\$ is the amplitude, \$k\$ the wave‑number and \$\omega\$ the angular frequency.
• The CRO displays a voltage signal as a function of time on its screen. By calibrating the horizontal (time) and vertical (voltage) scales, quantitative measurements can be extracted.
• Time‑base control sets the sweep speed \$S\$ (seconds per division). Y‑gain control sets the voltage per division \$V_{\text{div}}\$ (volts per division).

3. CRO Controls Relevant to Wave Measurements

4. Determining Frequency from the Time‑Base

When a sinusoidal voltage of frequency \$f\$ is applied to the CRO input, the screen shows a repeating pattern with a period \$T=1/f\$.

1. Measure the horizontal length \$L_{\text{h}}\$ (in divisions) of one complete cycle on the screen.
2. Calculate the period using the sweep speed:

   \$T = L_{\text{h}}\times S\$

3. Obtain the frequency:

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

5. Determining Amplitude from the Y‑Gain

The vertical peak‑to‑peak height \$L{\text{v}}\$ (in divisions) corresponds to the voltage peak‑to‑peak \$V{\text{pp}}\$ of the signal.

1. Measure \$L_{\text{v}}\$ on the screen.
2. Convert to voltage:

   \$V{\text{pp}} = L{\text{v}}\times V_{\text{div}}\$

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

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

6. Worked Example

Suppose a CRO is set to a time‑base of \$S = 0.5\ \text{ms/div}\$ and a y‑gain of \$V_{\text{div}} = 2\ \text{V/div}\$. The displayed waveform has a horizontal length of \$4\$ divisions per cycle and a vertical peak‑to‑peak height of \$3\$ divisions.

1. Period:

   \$T = 4\ \text{div}\times 0.5\ \text{ms/div}=2\ \text{ms}\$

2. Frequency:

   \$f = \frac{1}{2\ \text{ms}} = 500\ \text{Hz}\$

3. Peak‑to‑peak voltage:

   \$V_{\text{pp}} = 3\ \text{div}\times 2\ \text{V/div}=6\ \text{V}\$

4. Amplitude:

   \$A = \frac{6\ \text{V}}{2}=3\ \text{V}\$

7. Practical Tips

• Ensure the CRO is correctly calibrated; use the built‑in calibration signal if available.
• Adjust the time‑base so that at least one full cycle fits comfortably on the screen (typically 2–4 divisions).
• Set the y‑gain so that the waveform occupies a reasonable vertical range without clipping (about 2–4 divisions peak‑to‑peak).
• Use the CRO’s cursors or the built‑in measurement functions for more precise division counts.

8. Summary Table

9. Extension to Progressive Waves on a String

When a mechanical wave on a stretched string is transduced into an electrical signal (e.g., via a pickup coil), the same CRO techniques apply. The measured frequency \$f\$ corresponds to the wave’s temporal frequency, while the amplitude \$A\$ reflects the string’s transverse displacement converted to voltage.

By varying the driving frequency and observing the CRO trace, resonance conditions (maximal amplitude) can be identified, linking the CRO measurements directly to the physical properties of the progressive wave.