Lecture 11 WAVE

Electromagnetic waves The wave is a propagation of the disturbance through the medium without any net displacement of the medium Mechanical waves Electromagnetic waves traveling through empty space the benefit of a medium

Longitudinal wave A longitudinal wave is a wave in which the particles of the medium oscillate in simple harmonic motion parallel to the direction of the wave propagation

Transverse wave A transverse wave is a wave in which the particles of the medium execute simple harmonic motion in a direction perpendicular to its direction of propagation

Characteristics of a wave 1.The displacement of any particle of the wave is the displacement of that particle from its equilibrium position and is measured by the vertical distance y. 2. The amplitude of the wave is the maximum value of the displacement and denoted by A. 3. The wavelength of a wave is the distance, in the direction of propagation, in which the wave repeats itself and is denoted by  4. The period T of a wave is the time it takes for one complete wave to pass a particular point 5.The frequency f of a wave is defined as the number of waves passing a particular point per second a fundamental equation of wave propagation The wave causes a transfer of energy from one point in the medium to another point in the medium without the actual transfer of matter between these points

Mathematical representation of a wave The wave number k is the number of waves contained in the interval of 2

Mathematical representation of a wave The wave is periodic in both space and time. The space period is represented by the wavelength , and the time by the period T. a wave traveling to the right a wave traveling to the left

Mathematical representation of a wave

The speed of a transverse wave on a string Tension, not period! The speed of a transverse wave in a string depends on the tension T in the string and the mass per unit length of the string m/l

Reflection and transmission of a wave at the boundary of two different media Case 1: the wave goes from the less dense medium to the more dense medium The pulse slows down in going from the less dense medium to the more dense medium When a wave goes from a less medium to a more dense medium, the wavelength of the transmitted wave is less than the wavelength of the incident wave

Case 1: the wave goes from the more dense medium to the less dense medium When a wave goes from a more dense medium to a less dense medium, the transmitted wave moves faster than the incident wave and has a longer wavelength

The principle of superposition The principle of superposition states that whenever two or more wave disturbances pass a particular point in a medium, the resultant displacement of the point of the medium is the sum of the displacements of each individual wave disturbance. Two waves are in phase if they reach their maximum amplitudes at the same time, are zero at the same time, and have their minimum amplitudes at the same time The angle  is called the phase angle and a measure of how far wave 2 is displaced in the horizontal from wave 1. the wave displaced to the right the wave displaced to the left

The principle of superposition When two waves are in phase with each other, and waves are said to exhibit constructive interference

The principle of superposition If the waves are 180º out of phase the resultant wave is zero everywhere. This is called destructive interference.

The principle of superposition In general,

Standing waves – the vibrating string This is the equation of a standing wave or a stationary wave the waves are moving to the left and the right, but the resultant wave does not travel at all, it is a standing wave on a string

Standing waves – the vibrating string These points x = λ/2, λ, 3λ/2,2, where the amplitude of the standing wave is zero, called nodes These points x = λ/4, 3λ/4,5/4, where the amplitude of the standing wave is maximum, called antinodes

Standing waves – the vibrating string How many different types of standing waves can be produced on this string? The only wavelengths that are allowed on the string are =2L,L,2L/3, and so forth

Standing waves – the vibrating string tension n=1,2,3, …. the fundamental frequency n=1 n=2 the first overtone or second harmonic natural frequencies If we apply a periodic force to the string at any one of these natural frequencies, the mode of vibration continues as long as the driving force is continued. Such vibration is called a forced vibration.

The transmission of Energy in a wave and the intensity of a wave The total energy transmitted by the wave is equal to the total energy of the vibrating particle The total energy possessed by a single particle in simple harmonic motion is The intensity of a wave is defined as the energy of the wave that passes a unit area in a unit time The intensity of a mechanical wave of frequency f and amplitude A, moving at a velocity v in a medium of density 

Sound waves A sound wave is a longitudinal wave, this is, a particle of the medium executes simple harmonic motion in a direction that is parallel to the velocity of propagation. The speed of sound in a solid, where Y is Young modulus (N/m2),  is the density of the medium The speed of sound in a fluid, B is the bulk modulus (N/m2) The speed of sound in a gas,  is a constant called the ratio of the specific heats of the gas (1.40 for air), p is the pressure of the gas where t is the temperature of the air in degrees Celsius the loudness of a sound as heard by the human ear is characterized by the intensity level of a sound wave, measured in decibels (dB), is defined as where Io is the lowest threshold of hearing – The highest threshold of hearing, that is the threshold of pain -