# Calculating Wavelength and Expressions for Standing Waves | Homework Solution

• ee1215
In summary, a standing wave with frequency 25Hz and wavelength 1m has been created on a string of length 1.5m. The maximum amplitude of the wave is 5cm and there are 3 antinodes. The expressions for the wave traveling to the right and reflected wave traveling left are y = 0.025cos(157t - 6.28x) and y = 0.025cos(157t + 6.28x), respectively.
ee1215

## Homework Statement

A standing wave has been created on a string of length 1.5m. The frequency of the wave is 25Hz. The ends of the string are nodal points and there are 3 antinodes.
A)What is the wavelength of the standing wave?
B)If the maximum amplitude of the wave is 5cm, write complete expressions, including values for k and w, for the generated wave traveling to the right and the reflected wave traveling left. Take right as the +xˆ direction. (Hint: Remember the standing wave is the sum of two waves of equal amplitudes traveling in opposite directions.)

omega=2pi*f
k=(2pi)/f

## The Attempt at a Solution

A) L=(3*lambda)/2
so the wavelength = 1m

B) A=2.5cm = .025m
omega=2pi*f = 2pi(25) = 157 rad/s
k=(2pi)/h = 2pi/1 = 6.28m^-1

So in the right direction: y = 0.025cos(157t - 6.28x)
and the wave to the left is y = 0.025cos(157t + 6.28x)

Are these correct?

It looks all right, good job.

ehild

## What are standing waves?

Standing waves are a type of wave that occurs when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other. This results in a wave pattern that appears to be standing still, hence the name "standing waves".

## How are standing waves formed?

Standing waves are formed when two waves with the same frequency and amplitude, but traveling in opposite directions, interfere with each other. This can happen when a wave is reflected back on itself, or when two waves are traveling in opposite directions in the same medium.

## What is the difference between standing waves and traveling waves?

The main difference between standing waves and traveling waves is that standing waves appear to be standing still, while traveling waves move through a medium. Additionally, standing waves are formed by the interference of two waves, while traveling waves are created by a source that continuously produces waves.

## What are the characteristics of standing waves?

Standing waves have specific characteristics that distinguish them from other types of waves. These include nodes (points of zero amplitude) and antinodes (points of maximum amplitude). The distance between two adjacent nodes (or antinodes) is always half the wavelength of the standing wave.

## What are some real-world applications of standing waves?

Standing waves have many practical applications in fields such as music, engineering, and telecommunications. In music, standing waves are used to create certain musical notes and harmonics on string instruments. In engineering, standing waves are used to test the stability of structures and to measure the properties of materials. In telecommunications, standing waves are used to transmit and receive signals in antennas and other devices.

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