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    5-Impedance of The String-02-03-2023

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    efewfwedawas
    1. An exam will be held on Friday, October 3rd for 25 marks on Module 1. Questions will be converted to 5 marks for a digital assignment. Students should bring writing materials, a calculator, and mobile phones are prohibited. The exam will last 45 minutes. 2. A wave propagating on a string can be represented by the displacement equation. The velocity and acceleration of the wave can be determined from this equation and show the wave undergoes simple harmonic motion. 3. Impedance is a measure of the opposition to wave motion by a medium. For a transverse wave on a string, the characteristic impedance depends on factors like tension, density, and wave velocity.

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    5-Impedance of The String-02-03-2023

    Uploaded by

    efewfwedawas
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    1. An exam will be held on Friday, October 3rd for 25 marks on Module 1. Questions will be converted to 5 marks for a digital assignment. Students should bring writing materials, a calculator, and mobile phones are prohibited. The exam will last 45 minutes. 2. A wave propagating on a string can be represented by the displacement equation. The velocity and acceleration of the wave can be determined from this equation and show the wave undergoes simple harmonic motion. 3. Impedance is a measure of the opposition to wave motion by a medium. For a transverse wave on a string, the characteristic impedance depends on factors like tension, density, and wave velocity.

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    5-Impedance of the string-02-03-2023

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    1. An exam will be held on Friday, October 3rd for 25 marks on Module 1. Questions will be converted to 5 marks for a digital assignment. Students should bring writing materials, a calculator, and mobile phones are prohibited. The exam will last 45 minutes. 2. A wave propagating on a string can be represented by the displacement equation. The velocity and acceleration of the wave can be determined from this equation and show the wave undergoes simple harmonic motion. 3. Impedance is a measure of the opposition to wave motion by a medium. For a transverse wave on a string, the characteristic impedance depends on factors like tension, density, and wave velocity.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    10 views6 pages

    5-Impedance of The String-02-03-2023

    Uploaded by

    efewfwedawas
    1. An exam will be held on Friday, October 3rd for 25 marks on Module 1. Questions will be converted to 5 marks for a digital assignment. Students should bring writing materials, a calculator, and mobile phones are prohibited. The exam will last 45 minutes. 2. A wave propagating on a string can be represented by the displacement equation. The velocity and acceleration of the wave can be determined from this equation and show the wave undergoes simple harmonic motion. 3. Impedance is a measure of the opposition to wave motion by a medium. For a transverse wave on a string, the characteristic impedance depends on factors like tension, density, and wave velocity.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 6

    Announcement

    Hello all, on 10.03.2023 (Friday), you will have Exam for 25 marks. All the
    questions will be asked from Module 1. The 25 marks will be converted into 5 and
    consider for Digital Assignment 1. Please bring paper, pen, pencil, eraser,
    sharpener and calculator for the exam. Mobile phone is not allowed. Exam will be
    conducted for 45 minutes
    Dispersion Relation and Phase Velocity

    𝒚 𝒙, 𝒕 = 𝑨 𝒔𝒊𝒏 𝒌𝒙 − 𝝎𝒕

    Representation of periodic displacement of a system moving away from origin in x-


    axis with respect to time t.
    𝝏𝒚 𝒙,𝒕
    Wave velocity of the wave (𝝑𝒚 ) = = 𝑨 𝝎 𝐜𝐨𝐬 (𝒌𝒙 − 𝝎𝒕)
    𝝏𝒕

    𝝏𝟐 𝒚 𝒙,𝒕
    Acceleration of the wave (𝒂𝒚 ) = = −𝑨 𝝎𝟐 𝐬𝐢𝐧 𝒌𝒙 − 𝝎𝒕 = −𝝎𝟐 𝒚 𝒙, 𝒕
    𝝏𝒕𝟐

    Acceleration of the wave disturbance is equal to −𝝎𝟐 times its displacement,


    which indicates that the disturbance is simple harmonic motion.
    Dispersion Relation and Phase Velocity

    𝒚 𝒙, 𝒕 = 𝑨 𝒔𝒊𝒏 𝒌𝒙 − 𝝎𝒕

    Double differentiation with respect to x is

    𝝏𝟐 𝒚 𝒙, 𝒕 𝟐 𝐬𝐢𝐧 𝒌𝒙 − 𝝎𝒕 = −𝒌𝟐 𝒚 𝒙, 𝒕
    = −𝑨 𝐤
    𝝏𝒙𝟐
    𝝏𝟐 𝒚 𝒙, 𝒕
    𝟐 𝝎𝟐
    𝝏𝒕 = = 𝝑𝟐
    𝝏𝟐 𝒚 𝒙, 𝒕 𝒌𝟐
    𝝏𝒙𝟐
    𝝎
    =𝝑
    𝒌
    𝝑 is the velocity of the wave
    Impedance of a String
    The opposition to the wave motion offered by a medium when a wave propagates
    through it. The impedance offered by the string to the transverse wave traveling through
    it is called the characteristics impedance. It is denoted by Z.

    𝒚 𝒙, 𝒕 = 𝑨 𝒔𝒊𝒏 𝒌𝒙 − 𝝎𝒕 + 𝝋
    𝑻𝑨 𝐤 𝐜𝐨𝐬 (𝒌𝒙 − 𝝎𝒕 + 𝝋)
    𝝏𝒚 𝒁=
    𝑭𝒚 = 𝑻 = 𝑻𝑨 𝐤 𝐜𝐨𝐬 (𝒌𝒙 − 𝝎𝒕 + 𝝋) 𝑨 𝝎 𝐜𝐨𝐬 (𝒌𝒙 − 𝝎𝒕 + 𝝋)
    𝝏𝒙
    𝑻𝒌 𝑻 𝝎
    𝒁= = = 𝝆𝝑 =𝝑
    𝝏𝒚 𝒙, 𝒕 𝒌
    = 𝑨 𝝎 𝐜𝐨𝐬 (𝒌𝒙 − 𝝎𝒕 + 𝝋) 𝝎 𝝑
    𝝏𝒕
    𝝆 is the density of the medium in which the wave propagates, 𝝑 is the velocity of the wave,
    T is the tension experienced by the string, Z is the impedance experienced by the wave from the medium .
    Reflection of Waves at a Boundary

    Wave pulse traveling on a string

    Reflection from a HARD boundary


    At a fixed hard boundary, the displacement remains zero and
    the reflected wave changes its polarity (undergoes a 180°
    phase change). According to Newton's third law, the wall must
    be exerting an equal downward force and the force creates a
    wave pulse that propagates from right to left with opposite
    polarity

    Reflection from a SOFT boundary


    At a free soft boundary, the restoring force is zero, since
    the net vertical force at the free end must be zero. and the
    reflected wave has the same polarity (no phase change) as
    the incident wave.
    Transmission of Waves at a Boundary

    From high speed to low speed (low density to high density)

    From low speed to high speed (high density to low density)

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