Modern Physics

# Mechanical waves: characteristics, properties, formulas, types

A mechanical wave is a disturbance that needs a physical medium to spread. The closest example is sound, capable of being transmitted through a gas, liquid or solid.

Other known mechanical waves are those that occur when the tight string of a musical instrument is pressed. Or the typically circular ripples caused by a stone thrown into a lake.

The disturbance travels through the medium, producing various displacements in the particles that make it up, depending on the type of wave. As the wave passes, each particle in the middle performs repetitive motions that briefly separate it from its equilibrium position.

The duration of the disturbance depends on your energy. In wave motion, energy is what propagates from one side of the environment to the other, since the vibrating particles never move very far from their place of origin.

The wave and the energy it carries can travel great distances. When the wave disappears, its energy has dissipated in the middle, leaving everything as calm and quiet as before the disturbance.

## Types of mechanical waves

Mechanical waves are classified into three main groups:

– transverse waves.

– longitudinal waves.

– surface waves.

### transverse waves

In transverse waves, particles move perpendicular to the direction of propagation. For example, the particles in the sequence in the following figure oscillate vertically as the wave moves from left to right:

### longitudinal waves

In longitudinal waves, the propagation direction and the direction of movement of particles are parallel.

### surface waves

A sea wave combines longitudinal waves and transverse waves on the surface, so they are surface waves, traveling across the boundary between two different media: water and air, as shown in the figure below.

When breaking waves on the coast, the longitudinal components predominate. Therefore, it is observed that the algae close to the coast have an alternative movement.

### Examples of different types of waves: seismic movements

During earthquakes, there are several types of waves that move across the globe, including longitudinal and transverse waves.

Longitudinal seismic waves are called P waves, while transverse waves are S waves.

The name P is due to the fact that they are pressure waves and are also primary when they arrive first, while the transversal ones are S by “shear” or shear and are also secondary, as they arrive after P.

## Characteristics and properties

The yellow waves in Figure 2 are periodic waves, which consist of identical disturbances moving from left to right. Note that both a and b have the same value in each of the wave regions.

The periodic wave perturbations are repeated in time and space, adopting the shape of a sinusoidal curve characterized by having crests or peaks, which are the highest points, and valleys, where the lowest points are.

This example will be used to study the most important characteristics of mechanical waves.

### wavelength and wavelength

Assuming the wave in Figure 2 represents a vibrating string, the black line serves as a reference and divides the wave train into two symmetrical parts. This line will coincide with the position where the string is at rest.

The value of a is called the wave amplitude and is usually indicated by the letter A. In turn, the distance between two successive valleys or crests is the wavelength 1 and corresponds to the magnitude called b in Figure 2.

### Period and frequency

Being a repetitive phenomenon in time, the wave has a period T, which is the time necessary to execute a complete cycle, while the frequency f is inverse or reciprocal of the period and corresponds to the number of cycles performed per unit of time.

The frequency f has as units in the International System the inverse of time: s -1 or Hertz, in honor of Heinrich Hertz, who discovered radio waves in 1886. 1 Hz is interpreted as the frequency equivalent to one cycle or vibration per second.

The velocity v of the wave relates the frequency to the wavelength:

v = λ.f = l / T

### angular frequency

Another useful concept is the angular frequency ω given by:

ω = 2πf

The speed of mechanical waves is different depending on the medium in which they travel. As a general rule, mechanical waves are faster when traveling through a solid and slower in gases, including the atmosphere.

In general, the velocity of many types of mechanical waves is calculated by the following expression:

For example, for a wave that propagates along a sequence, the velocity is given by:

Tension in the string tends to return it to its balanced position, while mass density prevents this from happening immediately.

## Formulas and Equations

The following equations are useful for solving the following exercises:

Angular frequency:

ω = 2πf

Time course:

T = 1 / f

Linear mass density:

v = λ.f

v = λ / T

v = λ / 2π

Velocity of the wave that propagates in a string:

## Solved Examples

### Exercise 1

The sine wave shown in Figure 2 moves in the direction of the positive x-axis and has a frequency of 18.0 Hz. It is known that 2a = 8.26 cm and b / 2 = 5.20 cm. To locate:

a) Amplitude.

b) Wavelength.

c) Period.

d) wave velocity.

### Solution

a) The amplitude is a = 8.26 cm / 2 = 4.13 cm

b) The wavelength is l = b = 2 x20 cm = 10.4 cm.

c) The period T is the inverse of the frequency, therefore T = 1 / 18.0 Hz = 0.056 s.

d) The wave velocity is v = lf = 10.4 cm. 18 Hz = 187.2 cm/sec.

### Exercise 2

A 75 cm long thin wire has a mass of 16.5 g. One end is fixed to the rod, while the other has a screw that allows you to adjust the tension in the wire. Calculate:

a) The velocity of this wave.

b) The voltage in Newton necessary for a transverse wave with a wavelength of 3.33 cm to vibrate at a rate of 625 cycles per second.

### Solution

a) Using v = λ.f, valid for any mechanical wave and substituting numerical values, you get:

v = 3.33 cm x 625 cycles / second = 2081.3 cm / s = 20.8 m / s

b) The speed of the wave propagating through a string is:

The tension T in the string is obtained by squaring it on both sides of the equality and clearing:

T = v 2 .μ = 20.8 2 . 2.2 x 10 -6 N = 9.52 x 10 -4 N.

## The sound: a longitudinal wave

Sound is a longitudinal wave, very easy to visualize. For this, all that is needed is a stealth , a flexible coil spring with which many experiments can be carried out to determine the shape of the waves.

A longitudinal wave consists of a pulse that compresses and expands the medium in turn. The compressed area is called the “compression” and the area where the coils of the spring are further apart is the “expansion” or “rarefaction”. Both zones move along the axial axis of the slinky and form a longitudinal wave.

In the same way that one part of the dock is compressed and the other extends as the energy moves along the wave, the sound compresses parts of the air around the source that gives off the disturbance. For this reason, it cannot be spread in a vacuum.

For longitudinal waves, the parameters described above for periodic transverse waves are equally valid: amplitude, wavelength, period, frequency and wave velocity.

Figure 5 shows the wavelength of a longitudinal wave traveling along a coil spring.

In it, two points located at the center of two successive compressions were selected to indicate the wavelength value.

Compressions are equivalent to crests and expansions are from valleys in a transverse wave; therefore, a sound wave can also be represented by a sine wave.

## Sound characteristics: frequency and intensity

Sound is a type of mechanical wave with several very special properties that distinguish it from the examples we’ve seen so far. Next, we’ll see what its most relevant properties are.

### Frequency

The frequency of sound is perceived by the human ear as a high-pitched (high-frequency) or bass (low-frequency) sound.

The frequency range audible in the human ear is between 20 and 20,000 Hz. Above 20,000 Hz are the so-called ultrasound sounds and below the infra-sound, frequencies inaudible to humans, but that dogs and other animals can perceive and use

For example, bats emit ultrasound waves with their nose to determine their location in the dark and also through communication.

These animals have sensors with which they receive the reflected waves and, somehow, interpret the delay time between the emitted and reflected waves and the differences in their frequency and intensity. With these data, they infer the distance they traveled and, in this way, are able to know where the insects are and fly between the cracks of the caves they inhabit.

Marine mammals such as whales and dolphins have a similar system: they have specialized fat-filled organs in their heads, with which they emit sounds, and corresponding sensors in their jaws, which detect the reflected sound. This system is known as echolocation.

### Intensity

Sound wave intensity is defined as the energy transported per unit of time and per unit of area. Energy per unit of time is energy. Therefore, sound intensity is energy per unit area and comes in watts / m 2 or W / m 2 . The human ear perceives wave intensity as volume: the louder the music, the louder it will be.

The ear detects intensities between 10-12 and 1 W / m 2 without feeling pain, but the relationship between intensity and perceived volume is not linear. To produce a sound with twice the volume, a wave with 10 times the intensity is needed.

The sound intensity level is a relative intensity measured on a logarithmic scale, where the unit is the bel and, most often, the decibel or decibel.

The sound intensity level is indicated as β and is given in decibels by:

β = 10 log (I / I or )

Where I is the sound intensity and I or is a reference level that is considered the threshold of hearing at 1 x 10-12 W/m 2 .

## Practical experiences for children

Children can learn a lot about mechanical waves while having fun. Here are several simple experiments to see how waves transmit energy that can be harnessed.

### -Experiment 1: Intercom

#### Materials

– 2 plastic cups whose height is much greater than the diameter.

– Between 5 and 10 meters of strong wire.

#### Implementation

Pierce the base of the pots to pass the thread through them and secure it with a knot at each end so the thread does not come out.

– Each player takes a cup and walks away in a straight line, ensuring the string is taut.

– One of the players uses the glass as a microphone and talks to the partner, who, of course, must put the glass to his ear in order to hear. No need to scream.

The listener will immediately notice that the sound of their partner’s voice is transmitted through the tense wire. If the cord is not taut, your friend’s voice will not be heard clearly. Nor will you hear anything if the wire is placed directly in the ear, it is necessary to listen to the cup.

#### Explanation

We know from the previous sections that tension in the string affects the velocity of the wave. Transmission also depends on the material and diameter of the vessels. When the partner speaks, the energy of your voice is transmitted to the air (longitudinal wave), from there to the bottom of the vessel and then as a transverse wave through the wire.

The wire transmits the wave to the bottom of the listener’s vessel, which vibrates. This vibration is transmitted to the air and is sensed by the eardrum and interpreted by the brain.

### Experiment 2: Watching the waves

#### Implementation

On a table or flat surface is a sticky , flexible coil spring with which they can form various types of waves.

#### longitudinal waves

The ends are held, one in each hand. Then a small horizontal impulse is applied to one end and observed as a pulse propagates along the spring.

You can also place one end of the fixed stealth on some support or ask a partner to hold it by stretching it enough. That way, there is more time to observe how the compressions and expansions occur, spreading from one end of the pier to the other quickly, as described in the previous sections.

#### transverse waves

The slinky is also held by one end, stretching it sufficiently. The free end is slightly shaken by shaking it up and down. Note that the sinusoidal pulse moves along the spring and returns.