1. صدا و ارتعاش در صنعت
جلسه دوم
محمد رضا منظم

2. مروري بر رياضيات

3. Real Numbers

4. Indices

5. Logarithms

6. The Binomial Theorem

7. Trigonometry

8. Trigonometry (continue..)

9. Radian Measure

10. Complex Numbers complex number has both real and imaginary component
Mathematical operation is similar to real number e.g.
Conjugate of is

11. Modulus (magnitude) and Argument (angle)

12. Polar Form of a Complex Number
The argument can be written:
Therefore

13. Polar Form of a Complex Number
Replacing
(Euler’s formula)

14. Polar Form of a Complex Number (continue)
Z can be written as:

15. Fourier Analysis (Joseph Fourier (1706-1790))
* Fourier series
It enables periodic functions to be represented by infinite series of sine and cosine terms.
for a function
Fourier series for the function is:

16. Fourier Analysis (Joseph Fourier (1706-1790))
Infinite Fourier Transform and inverse Fourier Transform

17. Fundamental concept Wave: Some type of wave: Water wave
Any moving form-some shape or pattern that travels along without carrying all the medium with it.
Some type of wave:
Water wave
Wave on string- musical instrument
Mexican wave
Wind causing wave
Heat wave
Electromagnetic wave
Sound wave ….

18. Velocity, Frequency and wave length
The velocity of a wave (c):
the speed at which its wave-form travels along, the speed of any labelled part of the disturbance.
The frequency of a wave (f):
The number of oscillation it makes in 1 second.
In 1 second the wave has travelled “c” metres so that “c” metres contains “f” cycles of the wave.
Hence in space one complete wave is c/f metres long (wavelength λ)
The time of one oscillation is the period (T)

19. How Waves Travel
A wave travels essentially because:
One piece of the medium disturbed by the wave disturbs the next piece of medium ahead and gives up the motion to it.
The waves are :
Longitudinal: The pieces of the medium oscillate in the same direction as the wave propagate.
Transverse: The pieces of the medium oscillate perpendicular to the direction propagation of the wave.
Some other types including, Shear and Bending

20. Mathematical Description of Harmonic Wave
The disturbance at x1 at time t1 is due to the disturbance at position x0 which occurred at time t0.

21. Mathematical (continue)
For harmonic waves ( plane wave):
ensures the wave repeats every wavelength

22. Mathematical (continue)
Complete representation of a plane wave:
To allow the wave to have any value:
If we put:

23. Mathematical (continue)
Alternative equation for plane wave:

24. Mathematical (continue)
Complex Representation

25. Examples (1)
If the displacement of the particles of the medium is described by:
What is the amplitude, frequency ,wavelength and wave number and what is the speed of the wave?

26. Examples (2) The pressure fluctuations in air are described by:
What is the amplitude, frequency, wavelength and the speed of the wave.

27. Examples (3) The pressure, p, is described by:
If the pressure amplitude is 0.01 pa and at t=0 , x=0 the value of p is pa find Ar and Ai.

28. Thanks for Listening