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Variations on the wave equation [PDF]
Mathematical Methods in the Applied Sciences, 2001 AbstractA lot of vibration processes in mathematical physics are described by the wave equation or by related equations and systems, and plenty of research has been done on this subject. The results and methods obtained thereby have been very important in other fields of application, and they still are. They also had and still have an immense influence
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1964
According to (1.24), the wave equation in R 1 is of the form $$ {u_{xx}} - \frac{1} {{{\gamma |2}}}{u_{tt}} = 0,\quad where\quad \gamma = const > 0 $$ (2.1) The symbol x denotes a real variable here. Introduction of a new time scale t = γt shows that we may restrict ourselves to the case γ = 1.
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According to (1.24), the wave equation in R 1 is of the form $$ {u_{xx}} - \frac{1} {{{\gamma |2}}}{u_{tt}} = 0,\quad where\quad \gamma = const > 0 $$ (2.1) The symbol x denotes a real variable here. Introduction of a new time scale t = γt shows that we may restrict ourselves to the case γ = 1.
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1998
This chapter treats the equation $$\frac{{{\partial ^2}u}}{{\partial {t^2}}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega ,t \in \left( {0,T} \right)$$ (1) which is called the wave equation and plays an important role in mathematical physics.
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This chapter treats the equation $$\frac{{{\partial ^2}u}}{{\partial {t^2}}}\left( {x,t} \right) - {\omega ^2}\Delta u\left( {x,t} \right) = f\left( {x,t} \right),x \in \Omega ,t \in \left( {0,T} \right)$$ (1) which is called the wave equation and plays an important role in mathematical physics.
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1977
We will start by deriving the wave equation for vibrations of a string in one dimension. The waves or vibration are described by giving h, the displacement of the string off the χ axis, as a function of distance along the χ axis (see Figure 2.1). If this were a water wave, we would interpret h as the height of the water rather than as the displacement ...
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We will start by deriving the wave equation for vibrations of a string in one dimension. The waves or vibration are described by giving h, the displacement of the string off the χ axis, as a function of distance along the χ axis (see Figure 2.1). If this were a water wave, we would interpret h as the height of the water rather than as the displacement ...
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1974
Many of the earlier sections of this book may be summarised by saying that they dealt with superposition properties of waves of various kinds. As we saw in section 1.2, if it is assumed that the equation of motion of the wave is linear, the amplitude of different waves may be added.
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Many of the earlier sections of this book may be summarised by saying that they dealt with superposition properties of waves of various kinds. As we saw in section 1.2, if it is assumed that the equation of motion of the wave is linear, the amplitude of different waves may be added.
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2015
There are many equations that describe wave behavior and the relationship between wave parameters, and you may well find such equations referred to as “the wave equation”. In this chapter, you can read about the most common form of the wave equation, which is a linear, second-order, homogeneous partial differential equation (the meaning of each of ...
Daniel Fleisch, Laura Kinnaman
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There are many equations that describe wave behavior and the relationship between wave parameters, and you may well find such equations referred to as “the wave equation”. In this chapter, you can read about the most common form of the wave equation, which is a linear, second-order, homogeneous partial differential equation (the meaning of each of ...
Daniel Fleisch, Laura Kinnaman
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2000
The term ‘wave’ is used to signify an effect which propagates in a medium. The medium can be an object such as a string, cable, beam, plate, shell or a physical space consisting of fluids, solids, etc. The most common observation of a wave is that of the motion of the surface of still water when a pebble is thrown in it. The ‘snap’ at the end of a whip
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The term ‘wave’ is used to signify an effect which propagates in a medium. The medium can be an object such as a string, cable, beam, plate, shell or a physical space consisting of fluids, solids, etc. The most common observation of a wave is that of the motion of the surface of still water when a pebble is thrown in it. The ‘snap’ at the end of a whip
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2009
So far in this book we have dealt with vibrations. These are processes that vary as functions of time. We were able to describe relevant types of vibrations with common differential equations. This chapter now focuses on waves, which are processes that vary with both time and space. Their mathematical description requires partial differential equations.
Jens Blauert, Ning Xiang
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So far in this book we have dealt with vibrations. These are processes that vary as functions of time. We were able to describe relevant types of vibrations with common differential equations. This chapter now focuses on waves, which are processes that vary with both time and space. Their mathematical description requires partial differential equations.
Jens Blauert, Ning Xiang
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