A sound wave has wavelength 4 m and travels through a medium with speed 500 m/s. What is the time period of this wave, in seconds?

Introduction / Context:
Waves are described by several related quantities, including wavelength, frequency, speed, and time period. The time period is the time taken for one complete oscillation of the wave at a fixed point, and it is closely related to the wavelength and speed. This question focuses on a simple numerical relationship between these quantities for a sound wave. Knowing how to link speed, wavelength, and time period is essential in acoustics and general wave mechanics.

Given Data / Assumptions:
• Wavelength of the sound wave lambda = 4 m. • Speed of the wave v = 500 m/s. • The wave is travelling in a uniform medium where speed remains constant. • We are asked to find the time period T in seconds.

Concept / Approach:
The fundamental relation linking wave speed v, wavelength lambda, and frequency f is v = f * lambda. The time period T is the reciprocal of the frequency, T = 1 / f. Combining these two relations gives T = lambda / v. This means that the time period equals the distance of one wavelength divided by the speed of the wave. By substituting the given values of wavelength and speed, we can compute the time period directly.

Step-by-Step Solution:
Step 1: Use the formula for time period in terms of wavelength and speed: T = lambda / v. Step 2: Substitute lambda = 4 m and v = 500 m/s. Step 3: Compute T = 4 / 500 s. Step 4: Simplify 4 / 500 = 0.008 s. Step 5: Therefore, the time period of the wave is 0.008 seconds.

Verification / Alternative check:
Another way is to first find the frequency. Using v = f * lambda, we get f = v / lambda = 500 / 4 = 125 Hz. The time period is the reciprocal of frequency, so T = 1 / f = 1 / 125 s. Performing the division 1 / 125 gives 0.008 s, which matches the earlier calculation. This confirms that our method and result are consistent and correct.

Why Other Options Are Wrong:
Option a (2000 s) and option c (1000 s): These values are extremely large and physically unrealistic for a sound wave with such a short wavelength and high speed. Option b (0.016 s): This is double the correct time period and would correspond to a frequency of about 62.5 Hz, which does not match v = 500 m/s and lambda = 4 m. Option e (0.004 s): This is half of the correct value and would require a wavelength of 2 m for the same speed, not 4 m.

Common Pitfalls:
Some learners mistakenly use T = v / lambda instead of T = lambda / v, which inverts the answer. Others may confuse time period with frequency and directly choose a value that would be more appropriate as frequency rather than time. It is also easy to make arithmetic errors when dealing with decimals. To avoid mistakes, always start from v = f * lambda, solve for the quantity you need, and carefully check units and decimal placement.

Final Answer:
The time period of the sound wave is 0.008 s.