Wave

Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets. As these wavelets spread outward, they form a new wavefront. For a spherical wavefront, the wavelets originate from a point source, and as they travel, they create spherical surfaces around the source. The radius of each spherical wavefront expands over time, and the new wavefront is tangent to the wavelets.

Step 1: A stationary wave results from the interference of two identical waves traveling in opposite directions.
Step 2: The phase difference ( ) between a node (where displacement is zero) and an adjacent antinode (where displacement is maximum) is given by: This indicates that the particle at an antinode is a quarter cycle ahead of the particle at the node. \bigskip

A stationary wave is formed when two progressive waves of the same frequency and amplitude, traveling in opposite directions, superpose on a stretched string. The equation of the two waves can be written as: and where: - is the amplitude,
- is the wave number ( , where is the wavelength),
- is the angular frequency,
- is the position along the string,
- is time.
Now, the resultant displacement of the string is the sum of these two waves:
Substitute the expressions for and :
Using the trigonometric identity for the sum of sines, we have:
This is the equation of a stationary wave on the stretched string, where:
- The term represents the spatial variation (the shape of the wave along the string),
- The term represents the time variation.
Identifying Nodes and Antinodes:
- The term determines the spatial positions where the displacement is zero. These points are called nodes.
- The term reaches its maximum value at points where , and these points are called antinodes.
The nodes occur when , which happens at:
where is an integer. These are the positions of the nodes.
Distance Between Two Successive Nodes:
The distance between two successive nodes is:
Since the wavelength , the distance between two successive nodes is:

Distance Between Two Successive Antinodes:
The antinodes occur where , and the distance between two successive antinodes is also .
Thus, the distance between two successive nodes or two successive antinodes in a stationary wave on a stretched string is .

The wavelength of a sound wave is related to its frequency and the speed of sound by the following equation: For the two tuning forks, the wavelengths are: The difference in wavelengths is: