Understanding the Speed of Sound Relation

The question asks us to find the correct relationship between the speed of sound (denoted by v), its wavelength (denoted by the Greek letter lambda, $\lambda$), and its time period (denoted by T).

Key Concepts Explained

Let's break down the terms involved:

• Speed of Sound (v): This is how fast a sound wave travels through a medium (like air, water, or solids). It's measured in units like meters per second (m/s).
• Wavelength ($\lambda$): This is the distance between two consecutive identical points on a wave. Think of it as the length of one complete cycle of the wave. It's measured in units of distance, like meters (m).
• Time Period (T): This is the time it takes for one complete wave cycle to pass a certain point. It's measured in units of time, like seconds (s).

Deriving the Wave Equation

We know the basic definition of speed:

Speed = Distance / Time

Now, consider one full wave cycle:

• The distance covered by the wave in one cycle is equal to its wavelength ($\lambda$).
• The time taken for this one cycle is equal to the time period (T).

Substituting these into the speed formula, we get:

v = $\lambda$ / T

Alternative Perspective: Using Frequency

Another important concept is frequency (f), which is the number of wave cycles passing a point per unit time. Frequency is the reciprocal of the time period:

$f = 1 / T$

The general formula relating wave speed, frequency, and wavelength is:

v = f × $\lambda$

If we substitute $f = 1 / T$ into this equation, we get:

v = (1 / T) × $\lambda$

Which simplifies to:

v = $\lambda$ / T

Both approaches lead to the same relationship.

Analyzing the Options

Let's look at the given options based on our derived formula:

• Option 1: v = $\lambda$ × T - Incorrect.
• Option 2: v = $\lambda$ / T - Correct. This matches our derived formula.
• Option 3: v = $\lambda$ + T - Incorrect. Speed is not the sum of wavelength and time period.
• Option 4: v = T / $\lambda$ - Incorrect. This is the inverse of the correct relationship.

Conclusion

The correct relationship between the speed of sound (v), wavelength ($\lambda$), and time period (T) is v = $\lambda$ / T.