Mechanical Waves: Characteristics, Properties, Formulas, Types

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

Other well-known mechanical waves are those produced when the taut string of a musical instrument is plucked. Or the typically circular ripples caused by a stone thrown into a pond.

Figure 1. The taut strings of a musical instrument vibrate with transverse waves. Source: Pixabay.

The disturbance travels through the medium producing various displacements in the particles that compose it, depending on the type of wave. As the wave passes, each particle in the medium makes repetitive movements that briefly separate it from its equilibrium position.

The duration of the disturbance depends on its energy. In wave motion, energy is what propagates from one side of the medium to the other, since vibrating particles never stray too far from their place of origin.

The wave and the energy it carries can travel great distances. When the wave disappears, it is because its energy ended up dissipating in the middle, leaving everything as calm and silent as it was before the disturbance.

Article index

  • one

    Types of mechanical waves

    • 1.1

      Transverse waves

    • 1.2

      Longitudinal waves

    • 1.3

      Surface waves

    • 1.4

      Examples of the different types of waves: seismic movements

  • two

    Characteristics and properties

    • 2.1

      Wave amplitude and wavelength

    • 2.2

      Period and frequency

    • 23

      Angular frequency

  • 3

    Formulas and equations

  • 4

    Worked Examples

    • 4.1

      Exercise 1

    • 4.2

      Solution

    • 4.3

      Exercise 2

    • 4.4

      Solution

  • 5

    Sound: a longitudinal wave

  • 6

    The characteristics of sound: frequency and intensity

    • 6.1

      Frequency

    • 6.2

      Intensity

  • 7

    Practical experiments for children

    • 7.1

      -Experiment 1: Intercom

    • 7.2

      -Experiment 2: Observing the waves

  • 8

    References

Types of mechanical waves

Mechanical waves are classified into three main main groups:

– Transverse waves.

– Longitudinal waves.

– Surface waves.

Transverse waves

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

Figure 2. Transverse wave in a string. The direction of wave propagation and the direction of motion of an individual particle are perpendicular. Source: Sharon Bewick [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]

Longitudinal waves

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

Figure 3. Longitudinal wave. Source: Polpol [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0)]

Surface waves

In a sea wave, longitudinal waves and transverse waves are combined on the surface, hence they are surface waves, traveling on the border between two different media: water and air, as can be seen in the following figure.

Figure 4. Ocean waves that combine longitudinal and transverse waves. Source: modified from Pixabay.

When breaking waves on the shore, the longitudinal components predominate. For this reason, it is observed that the algae near the shore have a back and forth movement.

Examples of the different types of waves: seismic movements

During earthquakes, various types of waves are produced that travel over the globe, including longitudinal waves and transverse waves.

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

The P designation is due to the fact that they are pressure waves and they are also primary when arriving in the first place, while the transverse ones are S for “shear” or shear and are also secondary, since they arrive after the P.

Characteristics and properties

The yellow waves in Figure 2 are periodic waves, consisting 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 perturbations of the periodic wave are repeated both in time and in space, adopting the form of a sinusoidal curve characterized by having peaks or peaks, which are the highest points, and valleys where the lowest points are.

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

Wave amplitude and wavelength

Assuming that 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 would coincide with the position in which the rope is at rest.

The value of a is called the amplitude of the wave and is usually denoted by the letter A. For its part, the distance between two valleys or two successive crests is the wavelength l 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 it takes to carry out a complete cycle, while the frequency f is the inverse or reciprocal of the period and corresponds to the number of cycles carried out 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 length of the wave:

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 have higher speeds when they travel through a solid, and they are slower in gases, including the atmosphere.

In general, the speed of many types of mechanical wave is calculated by the following expression:

For example, for a wave that travels along a chord, the velocity is given by:

The tension in the string tends to return the string to its equilibrium position, while the mass density prevents this from happening immediately.

Formulas and equations

The following equations are useful in solving the exercises that follow:

Angular frequency:

ω = 2πf

Period:

T = 1 / f

Mass linear density:

v = λ.f

v = λ / T

v = λ / 2π

Speed ​​of the wave propagating in a string:

Worked Examples

Exercise 1

The sine wave shown in Figure 2 travels 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. Meet:

a) Amplitude.

b) Wavelength.

c) Period.

d) Wave speed.

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 speed of the wave is v = lf = 10.4 cm. 18 Hz = 187.2 cm / s.

Exercise 2

A thin wire 75 cm long has a mass of 16.5 g. One of its ends is fixed to a nail, while the other has a screw that allows adjusting the tension in the wire. Calculate:

a) The speed of this wave.

b) The tension in newtons necessary for a transverse wave whose wavelength is 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, we obtain:

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 rope is obtained by raising it squared to both sides of the equality and solving:

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

Sound: a longitudinal wave

Sound is a longitudinal wave, very easy to visualize. To do this, you only need a slinky , a flexible helical spring with which many experiments can be carried out to determine the shape of the waves.

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

Figure 5. Longitudinal wave propagating along a helical spring. Source: self made.

In the same way as one part of the spring is compressed and the other is stretched as the energy moves along with the wave, the sound compresses portions of the air that surrounds the source of the disturbance. For that reason it cannot propagate in a vacuum.

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

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

In it, two points located in the center of two successive compressions have been selected to indicate the value of the wavelength.

The compressions are the equivalent of the crests and the expansions are the equivalent of the valleys in a transverse wave, hence a sound wave can also be represented by a sine wave.

The characteristics of sound: frequency and intensity

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

Frequency

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

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

For example, bats emit ultrasound waves from their noses to determine their location in the dark and also for communication.

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

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

Intensity

The intensity of the sound wave is defined as the energy transported per unit of time and per unit of area. Energy per unit time is power. Therefore the intensity of sound is the power per unit area and comes in watt / m 2 or W / m 2 . The human ear perceives the intensity of the wave 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 requires a wave with 10 times more intensity.

The level of sound intensity is a relative intensity that is measured on a logarithmic scale, in which the unit is the bel and more frequently the decibel or decibel.

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

β = 10 log (I / I o )

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

Practical experiments for children

Children can learn a lot about mechanical waves while having fun. Here are some simple experiments to see how waves transmit energy, which 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.

Put into practice

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

– Each player takes a glass and they walk away in a straight line, ensuring that the thread remains taut.

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

The listener will immediately notice that the sound of his partner’s voice is transmitted through the taut thread. If the thread is not taut, your friend’s voice will not be heard clearly. Neither will you hear anything if you put the thread directly into your ear, the glass is necessary to listen.

Explanation

We know from the previous sections that the tension in the string affects the speed of the wave. The transmission also depends on the material and diameter of the vessels. When the partner speaks, the energy of his voice is transmitted to the air (longitudinal wave), from there to the bottom of the glass and then as a transverse wave through the thread.

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

-Experiment 2: Observing the waves

Put into practice

A slinky , the flexible helical spring with which various types of wave can be formed , lies on a table or flat surface .

Figure 6. Helical spring to play with, known as a slinky. Source: Pixabay.

Longitudinal waves

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

You can also place one end of the slinky fixed to a support or ask a partner to hold it, stretching it enough. This way there is more time to observe how the compressions and expansions follow one another propagating from one end of the spring to the other quickly, as described in the previous sections.

Transverse waves

The slinky is also held by one of its ends, stretching it enough. The free end is given a slight shake by shaking it up and down. The sinusoidal pulse is observed to travel along the spring and back.

References

  1. Giancoli, D. (2006). Physics: Principles with applications. Sixth Edition. Prentice Hall. 308- 336.
  2. Hewitt, Paul. (2012). Conceptual Physical Science. Fifth Edition. Pearson. 239-244.
  3. Rex, A. (2011). Fundamentals of Physics. Pearson. 263-273.

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