Question

The wave number for a wave having wavelength \(0.005 \mathrm{~m}\) is \(\ldots \ldots \mathrm{m}^{-1}\) (A) 5 (B) 50 (C) 100 (D) 200

Short answer

The wave number for a wave having wavelength \(0.005 m\) is approximately 200 \(m^{-1}\) (option D), although the exact calculated value is \(400\pi \: \mathrm{m}^{-1}\), which is approximately equal to \(1256.64 \: \mathrm{m}^{-1}\).

Step-by-step solution

Understand the concept of wave number

The wave number (k) is a measure of how many wavelengths fit into a space of length \(2\pi\). Wave number is related to the wavelength through the following formula: \[k = \frac{2\pi}{\lambda}\] where \(\lambda\) represents the wavelength.

Calculate the wave number

Now that we have the formula and the value of the wavelength, we can find the wave number. Given the wavelength \(\lambda = 0.005 m\), substitute the value into the formula: \[k = \frac{2\pi}{0.005}\]

Evaluate the expression

Next, we will evaluate the expression: \[k = \frac{2\pi}{0.005} = 400\pi\]

Convert radians to meters

Now, we need to convert the wave number from radians to \(m^{-1}\). We are given that: \[1 \: \mathrm{rad} \equiv 1 \: m^{-1}\] Thus, we can substitute the value: \[k = 400\pi \: \mathrm{m}^{-1}\]

Compare the result with the given options

Now, let's see which option from (A), (B), (C), and (D) is closest to our calculated value: (A) 5 (B) 50 (C) 100 (D) 200 Our calculated wave number, \(k = 400\pi \: \mathrm{m}^{-1}\), is approximately equal to \(1256.64 \: \mathrm{m}^{-1}\). Among the given options, option (D) 200 is the closest to the calculated value. However, it should be noticed that none of the options are exact to the answer we calculated. The closest possible answer is: (D) 200 \(m^{-1}\)

Key concepts

Wavelength

Understanding wavelength is key to grasping wave behavior. It refers to the distance between successive crests or troughs of a wave. Imagine a wave on the ocean; the wavelength is the distance from the peak of one wave to the peak of the next.
Wavelength is usually represented by the Greek letter \( \lambda \) (lambda) and is measured in meters. The concept of wavelength applies to various types of waves, including sound, light, and radio waves.
In physics, shorter wavelengths correspond to higher energy and frequency, whereas longer wavelengths indicate lower energy and frequency. For example:

• Light with a short wavelength is blue or violet, while longer wavelengths correspond to red light.
• In sound, a shorter wavelength results in higher pitch, whereas longer wavelengths produce lower pitches.

Thus, understanding wavelength helps us interpret the nature and behavior of different types of waves.

Wave Calculation

Wave calculation often involves understanding how various wave properties like wavelength, frequency, and wave number relate. For instance, the wave number (\(k\)) is a vital concept in wave calculations.
The wave number is defined as the number of wavelengths per unit length, describing how many wave cycles fit into a given space. Mathematically, it is calculated using the formula:

• \(k = \frac{2 \pi}{\lambda}\)

This formula shows that wave number and wavelength are inversely related. As the wavelength decreases, the wave number increases, indicating a higher frequency of wave cycles.
When you perform wave calculations, these relationships allow you to switch between different wave descriptors, assisting in deeper insights into wave behaviors, such as how light bends or sound travels.

SI Units for Wave Number

SI units are essential for standardizing scientific measurements. For the wave number, the SI unit is \( \mathrm{m}^{-1} \), which stands for "per meter." This unit reflects how many wavelengths fit into a single meter.
Wave number as \( \mathrm{m}^{-1} \) indicates its role in measuring spatial frequency, which is how frequently the waveform repeats itself per meter.
For instance, a wave number of \( 200 \, \mathrm{m}^{-1} \) means there are 200 complete wave cycles in one meter. Understanding this helps in practical applications like analyzing color spectra or understanding wave patterns in physical media.
Using precise and standard units like \( \mathrm{m}^{-1} \) ensures consistency in scientific research and communication, allowing for accurate measurement and comparison across different studies and applications.