Mechanical and Electromagnetic Vibrations and Waves 1st. Edition by Tamer Bécherrawy | 9781848212831 | 1848212836

Electromagnetics

Mechanical and Electromagnetic Vibrations and Waves

21 SOLVED PROBLEMS

Question: 7.3

Any incident wave on the interface of two dielectrics may be written in the form E = E// e// + E⊥e⊥, where e// and e⊥ are unit vectors that are perpendicular to the direction of propagation. e// is in the plane of incidence and e⊥ is perpendicular to this plane. a) Write in this form the expression ...

Verified Answer:

a) The wave may be written in the form

\un...

Question: 9.1

Determine the angular frequency of the first four modes of a square membrane of sides a if the speed of propagation is v. What is their degeneracy? Determine the nodal lines of the modes u12 and u21. Determine the nodal lines of the linear superposition u12 + u21 if they have the same amplitude ...

Verified Answer:

The normal frequencies are given by [9.22], i.e [l...

Question: 8.2

A radar station uses an emitter that consists of 30 rectilinear parallel wires separated by a distance d = 20 cm between consecutive wires. The station uses a wavelength of 10 cm. a) What are the directions of the principal maximums of order 0 and 1 and what is their angular width. b) Instead of ...

Verified Answer:

a) If the wave is emitted in a direction making an...

Question: 9.2

A string that is fixed at both ends is initially deformed by displacing its middle by a distance d before releasing it without an initial velocity (Figure 9.5). a) Write down the expression of the produced wave. b) Calculate the total energy of the excited modes and verify that it is equal to the ...

Verified Answer:

a) The initial conditions (for

t = 0 [/lat...

Question: 8.1

In a Young’s experiment the sources are two slits of width ε = 1.0 µm, at 0.2 mm apart. They are illuminated by a mercury lamp of wavelength λ = 546 nm. The interference pattern is observed on a screen in the focal plane of a converging lens of focal length f = 100 cm. a) Determine the spacing ...

Verified Answer:

a) The direction

\theta _p

of the...

Question: 7.2

Determine the energy flux densities of the incident, reflected and transmitted waves. Verify the conservation of energy at the interface. ...

Verified Answer:

The vector energy flux density of an acoustic wave...

Question: 5.2

Calculate the energy density, the energy flux density and the momentum density: a) for a progressive plane sound wave, whose velocity potential is Ψ=f(t–e.r/vs); and b) for a progressive spherical diverging sound wave whose velocity potential is Ψ=(R/r)Ψm cos (ωt-kr). Show that, at a large distance ...

Verified Answer:

a) Writing

\Psi =f(\xi ) \text { where } \...

Question: 1.2

a) Using the complex exponential representation, evaluate the sum of the simple harmonic functions u1 = A cos(ωt) and u2 = B sin(ωt). b) Derive the same result using the phasor diagram. ...

Verified Answer:

a) The functions

u_1

and

...

Question: 7.1

A membrane of mass per unit area ms separates two gases but it exerts no force of pressure on them (Figure 7.2). Write the boundary conditions on the membrane. Deduce the reflection coefficient and the transmission coefficient. Write down the expressions of the real waves of displacement in the ...

Verified Answer:

In this case, the displacement

\xi(t) [/la...

Question: 5.1

a) Evaluate the speed of sound in a mixture of two ideal gases of masses m1 and m2 occupying the unit volume. Their molar masses are M1 and M2 and their adiabatic constants are γ1 and γ2. b) Calculate the speed of sound at 20°C in air (21% of O2 and 79% of N2 in volume). MO = 32 and MN = 28. ...

Verified Answer:

a) The density of this mixture is

m_v=(m_1...