A Possible Generalization of Acoustic Wave Equation Using the Concept of Perturbed Derivative Order

Abstract

The standard version of acoustic wave equation is modified using the concept of the generalized Riemann-Liouville fractional order derivative. Some properties of the generalized Riemann-Liouville fractional derivative approximation are presented. Some theorems are generalized. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The numerical simulation of solution of the modified equation gives a better prediction than the standard one.

1. Introduction

Acoustics was in the beginning the study of small pressure waves in air which can be detected by the human ear: sound. The possibility of acoustics has been extended to higher and lower frequencies: ultrasound and infrasound. Structural vibrations are now often included in acoustics. Also the perception of sound is an area of acoustical research. In our present paper we will limit ourselves to the original definition and to the propagation in fluids like air and water. In such a case acoustics is a part of fluid dynamics. A major problem of fluid dynamics is that the equations of motion are nonlinear. This implies that an exact general solution of these equations is not available. Acoustics is a first-order approximation in which nonlinear effects are neglected. The corresponding relative density fluctuations 𝜌^′/𝜌₀ are considered very small [1]. The acoustic wave equation governs the propagation of acoustic waves through a material medium. The form of the equation is a second-order partial differential equation. The equation describes the evolution of acoustic pressure 𝑃 or particle velocity 𝑢 as a function of position 𝑟 and time 𝑡. A simplified form of the equation that describes acoustic waves in only one spatial dimension is considered in this paper 𝜕²𝑃𝜕𝑥²−1𝑐²𝜕²𝑃𝜕𝑡²=0. (1)

A derivation of general linearized wave equations is discussed by Pierce and Goldstein [1, 2]. However, neglecting the nonlinear effects in this equation, may lead to inaccurate prediction of the propagation of acoustic wave through the medium. Therefore in order to include explicitly the effect of corresponding relative density fluctuations in the mathematical formulation, one needs to insert it in the partial differential equation that governs the propagation of the acoustic wave. Recently, the acoustic equation was extended to the concept of fractional order derivative in [3]. Therefore in this paper, our concern is the modification of the previous equation by perturbing the order of the first derivative by replacing the first order of the derivative with 1+𝜀 where 𝜀 is a positive small parameter. Also, when we consider diffusion process in porous medium, if the medium structure or external field changes with time, in this situation, the ordinary integer order and constant-order fractional diffusion equation model cannot be used to well characterize such phenomenon (see [3–9]).

2. Definitions and Approximation

To describe the propagation of acoustic waves through a material medium with coordinate and time-dependent perturbed dimension, one must use Riemann-Liouville fractional order derivative that was introduced and used in a number of works (see [3, 6–8, 10]). These derivatives are defined as (see [3, 6–8]) 𝐷^1+𝜀𝑡=𝐷^𝜇𝑡_+,𝑡𝑓=𝑑𝑑𝑡^𝑛𝑡₀𝑓(𝜏)Γ𝑛−𝜇_𝜏(𝜏)(𝑡−𝜏)^{𝜇𝜏−𝑛+1}𝐷𝑑𝜏,_1+𝜀𝑥=𝐷^𝜇𝑥_+,𝑥𝑓=𝑑𝑑𝑥^𝑛𝑥₀𝑓(𝜏)Γ𝑛−𝜇_𝜏(𝜏)(𝑡−𝜏)^{𝜇𝜏−𝑛+1}𝑑𝜏. (2)

Here, Γ is the Euler gamma function; 𝑛={𝜇}+1, where {𝜇} is the integer part of 𝜇 for 𝜇≥0, that is, 𝑛−1≤𝜇<𝑛 and 𝑛=0 for 𝜇<𝑛. Following (2) we have that 𝜇_𝑡=1+𝜀_𝑡 and 𝜇_𝑥=1+𝜀_𝑥. The integral operator defined previously for fractional exponents 𝜇_𝑥and 𝜇_𝑡 depending on coordinates and time can be expressed in terms of ordinary derivative and integral [11] for |𝜀|≪1. Here 𝜀_𝑡 and 𝜀_𝑥 are considered as the corresponding relative density fluctuations 𝜌^′/𝜌₀ that vary slightly in time and space, respectively. For this matter, generalized Riemann-Liouville fractional derivatives satisfy the approximate relations: 𝐷^1+𝜀𝑡𝑓≅1+𝜀_𝑡𝜕𝑓+𝜕𝑡𝜕𝜀_𝑡𝐷𝜕𝑡𝑓,_1+𝜀𝑥𝑓≅1+𝜀_𝑥𝜕𝑓+𝜕𝑥𝜕𝜀_𝑥𝜕𝑥𝑓. (3)

The previous relations make it possible to describe the dynamic system, including the effect of the corresponding relative density fluctuations, by means of partial differential and integral equations.

3. Some Properties of the Approximation

Let us examine some properties of the previous derivative operator.

(i) Addition

If 𝑢_𝑥, 𝑓(𝑥), and 𝑔(𝑥) are differentiable in the opened interval I then, 𝐷^1+𝑢𝑥𝑓(𝑥)+𝑔(𝑥)≅𝐷^1+𝑢𝑥𝑓(𝑥)+𝐷^1+𝑢𝑥𝑔(𝑥). (4)

Proof. We have 𝐷^1+𝑢𝑥𝑓(𝑥)(𝑥≅+𝑔)1+𝑢_𝑥𝜕𝑓(𝑥)+𝑔(𝑥)+𝜕𝑥𝜕𝑢_𝑥𝑓(𝑥)(𝑥𝜕𝑥+𝑔)1+𝑢_𝑥𝜕𝑓(𝑥)+𝜕𝑥𝜕𝑢_𝑥𝑓(𝑥+𝜕𝑥)1+𝑢_𝑥𝜕𝑔(𝑥)+𝜕𝑥𝜕𝑢_𝑥𝑔(𝑥𝜕𝑥)≅𝐷^1+𝑢𝑥𝑓(𝑥)+𝐷^1+𝑢𝑥𝑓(𝑥). (5)

(ii) Division

If 𝑢_𝑥 and 1/𝑓(𝑥) are differentiable on the open interval I  then 𝐷^1+𝑢𝑥1𝑓(𝑥)≅−1+𝑢_𝑥𝑓^′(𝑥)+𝑢^′_𝑥𝑓(𝑥)𝑓²(𝑥)=−𝑓^′(𝑥)𝑓²(𝑥)−𝑢_𝑥𝑓^′(𝑥)𝑓²(𝑥)+𝑢^′_𝑥𝑓(𝑥)𝑓²(𝑥). (6)

(iii) Multiplication

If 𝑢_𝑥, 𝑓(𝑥) and 𝑔(𝑥) are differentiable in the open interval I then 𝐷^1+𝑢𝑥𝑓(𝑥)⋅𝑔(𝑥)≅𝑔(𝑥)𝑓^′(𝑥)+𝑓(𝑥)𝑔^′(𝑥)+𝑔𝑓^′+𝑓𝑔^′(𝑥)𝑢_𝑥+𝑢^′_𝑥𝑓(𝑥)𝑔(𝑥). (7)

(iv) Power

If 𝑢_𝑥 and 𝑓(𝑥) are differentiable in the open interval I then 𝐷^1+𝑢𝑥(𝑓(𝑥))^𝑛≅𝑛𝑓^′𝑓^𝑛−1+𝑢_𝑥𝑛𝑓^′𝑓^𝑛−1+𝑢^′_𝑥𝑓^𝑛,𝑛≥1. (8)

If 𝑢_𝑥 and 𝑓(𝑥) are two times differentiable in the open interval I then 𝐷^1+𝑢𝑥𝐷^1+𝑢𝑥𝑓(𝑥)≅1+𝑢_𝑥×1+𝑢_𝑥𝜕²𝑓𝜕𝑥²+3𝜕𝑓𝜕𝑥𝜕𝑢_𝑥+𝜕𝜕𝑥²𝑢_𝑥𝜕𝑥²𝑓+𝜕𝑢_𝑥𝜕𝑥𝑓. (9)

3.1. Clairaut’s Theorem for the Approximation

Assume that 𝑓(𝑥,𝑦), 𝑢_𝑥, and 𝑢_𝑦 are functions for which 𝜕²𝑓/𝜕𝑥𝜕𝑦, 𝜕²𝑓/𝜕𝑦𝜕𝑥,   𝜕²𝜀_𝑥/𝜕𝑥𝜕𝑦, and 𝜕²𝜀_𝑦/𝜕𝑥𝜕𝑦 exist and are continuous over a domain 𝐷⊂ℝ² then, 𝐷^1+𝑢𝑥[𝐷^1+𝑢𝑦[𝑓(𝑥,𝑦)]] and 𝐷^1+𝑢𝑦[𝐷^1+𝑢𝑥[𝑓(𝑥,𝑦)]] exist and are continuous over the domain D. If in addition 𝑢_𝑥=𝑢_𝑦 then 𝐷^1+𝑢𝑦𝐷^1+𝑢𝑥𝑓𝑥,𝑦=𝐷^1+𝑢𝑥𝐷^1+𝑢𝑦𝑓𝑥,𝑦. (10)

Proof. If 𝑓(𝑥,𝑦), 𝑢_𝑥, and 𝑢_𝑦 are functions for which 𝜕²𝑓/𝜕𝑥𝜕𝑦, 𝜕²𝑓/𝜕𝑦𝜕𝑥,𝜕²𝜀_𝑥/𝜕𝑥𝜕𝑦, and 𝜕²𝜀_𝑦/𝜕𝑥𝜕𝑦 exist and are continuous over a domain 𝐷⊂ℝ² then 𝐷^1+𝑢𝑦𝐷^1+𝑢𝑥𝑓≅𝑥,𝑦1+𝑢_𝑦𝜕𝑢_𝑥𝜕𝑦𝜕𝑓+𝜕𝜕𝑥²𝑓+𝜕𝜕𝑦𝜕𝑥²𝑢_𝑥𝑓+𝜕𝑦𝜕𝑥𝑥,𝑦𝜕𝑢_𝑥𝜕𝑥𝜕𝑓+𝜕𝑦1+𝑢_𝑥𝜕𝑢_𝑦𝜕𝑦𝜕𝑓+𝜕𝑥𝜕𝑢_𝑦𝜕𝑦𝜕𝑢_𝑥𝑓.𝜕𝑥𝑥,𝑦 (11) Now interchanging 𝑥 by 𝑦 we obtain 𝐷^1+𝑢𝑥[𝐷^1+𝑢𝑦[𝑓(𝑥,𝑦)]].
If 𝑢_𝑥 = 𝑢_𝑦 then 𝜕²𝑓/𝜕𝑦𝜕𝑥=𝜕²𝑓/𝜕𝑥𝜕𝑦 according to Clairaut’s theorem; thus replacing 𝜕²𝑓/𝜕𝑥𝜕𝑦 by 𝜕²𝑓/𝜕𝑦𝜕𝑥 in 𝐷^1+𝑢𝑦[𝐷^1+𝑢𝑥[𝑓(𝑥,𝑦)]], we obtain that 𝐷^1+𝑢𝑦𝐷^1+𝑢𝑥𝑓𝑥,𝑦=𝐷^1+𝑢𝑥𝐷^1+𝑢𝑦𝑓𝑥,𝑦. (12)

3.2. Chain-Rule for the Approximation

We have 𝐷^1+𝑢𝑥𝑓∘𝑔≅1+𝑢_𝑥𝑔^′(𝑥)𝑓^′𝑔(𝑥)+𝜕𝜀_𝑥𝜕𝑥𝑓∘𝑔=𝑔^′(𝑥)𝑓^′𝑔(𝑥)+𝑢_𝑥𝑔^′(𝑥)𝑓^′𝑔(𝑥+)𝜕𝜀_𝑥.𝜕𝑥𝑓∘𝑔 (13)

3.3. Rolle's Theorem for the Approximation

If a real-valued functions 𝑓 and 𝑢_𝑥 are continuous on a closed interval [a, b], differentiable on the open interval (𝑎,𝑏), and 𝑓(𝑎)=𝑓(𝑏), then there exist a 𝑐 in the open interval (𝑎,𝑏) and a small parameter 𝜇 such that 𝐷^1+𝑢𝑥𝑓(𝑐)=𝜇𝑓(𝑐). (14)

Proof. Following Rolle's theorem, there exists a c in the open interval (𝑎,𝑏)  such that 𝑓^′(𝑐)=0. For this 𝑐 we have that 𝐷^1+𝑢𝑥𝑓(𝑐)=1+𝑢_𝑥(𝑐)𝑓^′(𝑐)+𝑢^′_𝑥(𝑐)𝑓(𝑐)=𝑢^′_𝑥(𝑐)𝑓(𝑐)=𝜇𝑓(𝑐). (15)
If 𝑔(𝑥), 𝑓(𝑥), and 𝑢_𝑥 are differentiable in an open interval I, then there exist 2>𝛼>1 and 𝛽>0 such that 𝐷^1+𝑢𝑥𝑓(𝑥)−𝐷^1+𝑢𝑥𝑔(𝑥)≤𝛼𝑓^′(𝑥)−𝑔^′(𝑥𝑓(𝑥)(𝑥)+𝛽−𝑔),∀𝑥𝜖𝐼. (16)

Proof. Let 𝑥∈𝐼; then 𝐷^1+𝑢𝑥𝑓(𝑥)−𝐷^1+𝑢𝑥𝑔(𝑥=)1+𝑢_𝑥(𝑥)𝑓^′(𝑥)+𝑢^′_𝑥(𝑥)𝑓(𝑥)−1+𝑢_𝑥(𝑥)𝑔^′(𝑥)−𝑢^′_𝑥(𝑥)𝑔(𝑥=)1+𝑢_𝑥(𝑥)𝑓^′(𝑥)−𝑔^′(𝑥)+𝑢^′_𝑥(𝑥)𝑓(𝑥)−𝑔(𝑥). (17) But 𝑢_𝑥(𝑥) is very small such that |1+𝑢_𝑥(𝑥)|<2 and |𝑢^′_𝑥(𝑥)|>0; it follows that 1+𝑢_𝑥(𝑥)𝑓^′(𝑥)−𝑔^′(𝑥)+𝑢^′_𝑥(𝑥𝑓(𝑥)(𝑥𝑓)−𝑔)≤𝛼^′(𝑥)−𝑔^′(𝑥)+𝛽𝑓(𝑥)−𝑔(𝑥). (18)
It is important to observe that if 𝑢=0, we recover the properties of normal derivatives.

4. Modification of the Equation

In order to include explicitly the possible effect of the corresponding relative density fluctuations into the mathematical formulation, in this paper, we replace the classical version of the derivative of (1) by the modified Riemann-Liouville fractional derivative approximation (3) to obtain 𝐷^1+𝜀𝑥𝐷^1+𝜀𝑥𝑃=1𝑐²𝐷^1+𝜀𝑡𝐷^1+𝜀𝑡𝑃. (19) Making use of (3) and relation (8), the previous equation can be transformed to the following partial differential equation for 𝜀≪1: 1+𝜀_𝑥1+𝜀_𝑥𝜕²𝑃𝜕𝑥²+3𝜕𝑃𝜕𝑥𝜕𝜀_𝑥+𝜕𝜕𝑥²𝜀_𝑥𝜕𝑥²𝑃+𝜕𝜀_𝑥𝑃=1𝜕𝑥𝑐²1+𝜀_𝑡1+𝜀_𝑡𝜕²𝑃𝜕𝑡²+3𝜕𝑃𝜕𝑡𝜕𝜀_𝑡+𝜕𝜕𝑡²𝜀_𝑡𝜕𝑡²𝑃+𝜕𝜀_𝑡𝑃.𝜕𝑡 (20) Omitting the terms of 𝜀² in the previous equation, we obtain the following: 1+2𝜀_𝑥𝜕²𝑃𝜕𝑥²+3𝜕𝑃𝜕𝑥𝜕𝜀_𝑥+𝜕𝜕𝑥²𝜀_𝑥𝜕𝑥²𝑃=1𝑐²1+2𝜀_𝑡𝜕²𝑃𝜕𝑡²+3𝜕𝑃𝜕𝑡𝜕𝜀_𝑡+𝜕𝜕𝑡²𝜀_𝑡𝜕𝑡²𝑃. (21)

Now since the small parameters representing the perturbation additions to unity are small, the right- and left-hand sides of (21) can be divided by 1+2𝜀_𝑥. In this case, we obtain 𝑐()𝜕𝑥,𝑡²𝑃𝜕𝑡²=𝜕²𝑃𝜕𝑥²+𝐹(𝑥,𝑡)+𝐵(𝑥,𝑡)𝑃. (22) Here, We have, 𝑐()=1𝑥,𝑡𝑐²1+2𝜀_𝑡−2𝜀_𝑥,𝐹(𝑥,𝑡)=3𝜕𝑃𝜕𝑥𝜕𝜀_𝑥−1𝜕𝑥𝑐²𝜕𝑃𝜕𝑡𝜕𝜀_𝑡,𝐵()=𝜕𝜕𝑡𝑥,𝑡²𝜀_𝑥𝜕𝑥²−1𝑐²𝜕²𝜀_𝑡𝜕𝑡². (23)

It is easy to observe that (22) differs from (1) in three properties.

First the velocity of the sound in this case depends on time and coordinates due to the effect of the corresponding relative density fluctuations. Secondly the force 𝐹(𝑥,𝑡)=3𝜕𝑃𝜕𝑥𝜕𝜀_𝑥−1𝜕𝑥𝑐²𝜕𝑃𝜕𝑡𝜕𝜀_𝑡𝜕𝑡 (24) appears due to the coordinate and time dependence of the corresponding relative density fluctuations, and such force was considered in [12]. Third there is a derivative-free term that depends on both time and space, and it is proportional to 𝑃 and characterizes, depending on the coefficient sign, the retardation or enhancement propagation of acoustic waves through a material medium. Therefore, even weak memory, which is taken into account by generalized Riemann-Liouville fractional derivatives and presents the characteristics of a corresponding relative density fluctuations, transforms constant-coefficient velocity to varying-coefficient velocity. Moreover, this memory is responsible for a force with which the corresponding relative density fluctuations act on a propagation of acoustic waves through a material medium. This force appears only if the propagating acoustic wave has memory depending on coordinates and time; that it “remembers” its trajectories and time. Those terms in (22) that involve the corresponding relative density fluctuations additions (𝐹 and 𝐵) to the time and space dimensions are very small. Also the exact analytical solution of this modified equation is not easy to be determined. Therefore, this equation can be solved approximately by changing the function 𝑃(𝑟,𝑡) to 𝑃₀(𝑟,𝑡), which satisfies (1).

5. Solutions of the Modified Equation

The modified equation can be reformulated as follows: −1𝑐²𝜕²𝑃𝜕𝑡²+𝜕²𝑃𝜕𝑥²=𝐻𝜀_𝑥,𝜀_𝑡,𝑃₀(𝑟,𝑡). (25)

To solve (25) we need to give explicitly 𝜀_𝑥 and 𝜀_𝑡. For example if one consider these functions to be 𝜀_𝑥=𝜀_𝑜𝑥sin𝜃_𝑥𝑥,𝜀_𝑡=𝜀_𝑜𝑡sin𝜃_𝑡𝑡, (26) where 𝜀_𝑜𝑥 and 𝜀_𝑜𝑡 are very significantly small such that |𝜀|≪1. Since these parameters are very small then the solution of (25) can be sought in the form 𝑃=𝑃₁+𝑃₀ where 𝑃₀ is the solution of (1) and 𝑃₁≪𝑃₀ is proportional to 𝜀_𝑥 and 𝜀_𝑡. Here 𝜃_𝑥 and 𝜃_𝑡 are frequencies characterizing variation in corresponding relative density fluctuations.

To solve (25) together with (26) we make use of two techniques including the Green function techniques and the variational iterative decomposition technique. Here we will start with the variational iteration techniques.

5.1. Variational Iteration Method

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. Very recently it was recognized that the variational iteration method [11, 13–20] can be an effective procedure for solution of various nonlinear problems without usual restrictive assumptions. In this paper we will make use of this iterative decomposition technique to solve the modified wave (26) together with (25). To solve (25) by means of variational iteration method, we put (25) in the form of 𝑃₁(𝑥,𝑡)_2𝑥−1𝑐²𝑃₁(𝑥,𝑡)_2𝑡−𝐻(𝑥,𝑡)=0. (27)

The correction functional for (5) can be approximately expressed for this matter as follows: 𝑃_𝑛+1(𝑥,𝑡)=𝑃_𝑛()−𝑥,𝑡_𝑥0𝜆(𝜁𝜕)^𝑚𝑃(𝜁,𝑡)𝜕𝜁^𝑚−1𝑐²(𝑃(𝜁,𝑡))_2𝑡−𝐻(𝜁,𝑡)𝑑𝜁, (28) where 𝜆 is a general Lagrange multiplier [21, 22], which can be recognized optimally by means of variation assumption [21, 22]; here (𝑃(𝜁,𝑡))_2𝑡, and 𝐻(𝑟,𝜏) are considered as constrained variations. Making the previous functional stationary, we obtain 𝛿𝑃_𝑛+1(𝑥,𝑡)=𝛿𝑃_𝑛(𝑥,𝑡)−𝛿^𝑥₀𝜆(𝜁𝜕)^𝑚𝑃(𝜁,𝑡)𝜕𝜁^𝑚𝑑𝜁. (29)

Capitulates the next Lagrange multipliers, giving up to the following Lagrange multipliers 𝜆=−1 for the case where 𝑚=1 and 𝜆=𝑥−𝜁 for 𝑚=2. For these matters 𝑚=2, we obtain the following iteration formula: 𝑃_𝑛+1(𝑥,𝑡)=𝑃_𝑛(𝑥,𝑡)−^𝑥₀()×𝜕𝑥−𝜁₂𝑃_𝑛(𝜁,𝑡)𝜕𝜁²−1𝑐²𝜕²𝑃_𝑛(𝜁,𝑡)𝜕𝑡²−𝐻(𝜁,𝑡)𝑑𝜁. (30)

It is worth noting that if the zeroth component 𝑃₀(𝑥,𝑡) is defined, then the remaining components 𝑛≥1 can be completely determined such that each term is determined by using the previous terms, and the series solutions are thus entirely determined. Finally, the solution 𝑃(𝑟,𝑡) is approximated by the truncated series 𝑃_𝑁(𝑥,𝑡)=^𝑁−1_𝑛=0𝑃_𝑛(𝑥,𝑡),lim_𝑁→∞𝑃_𝑁(𝑟,𝑡)=𝑃₁(𝑥,𝑡). (31)

Here we choose the first term to be zero meaning 𝑃₀(𝑥,𝑡)=𝑃(0,𝑡)=0 and the second term can be determined as 𝑃₁(𝑥,𝑡)=−^𝑥₀(𝑥−𝜁)−𝐻(𝜁,𝑡)𝑑𝜁. (32)

Our next concern is to define 𝐻(𝑥,𝑡); that is, we first need to provide the solution of (1) which is found in the literatures [23]. The following solutions are obtained by separation of variables in different coordinate systems. They are phasor solutions, meaning that they have an implicit time-dependence factor of exp(𝑖𝜔𝑡) where 𝜔=2𝜋𝑓 is the angular frequency. The explicit time dependence is given by (33) 𝑃()𝑝()[𝜔𝑥,𝑘,𝑡=Real𝑘,𝑥exp𝑖𝜔𝑡],𝑘=𝑐. (33)

Introducing the previous expression in 𝐻(𝑥,𝑡), we obtain the second expression. In this matter two components of the decomposition series were obtained of which 𝑃(𝑥,𝑡) was evaluated to have the following expansion: 𝑃₁(𝑥,𝑡)=𝑃₁(𝑥,𝑡)+𝑃₀(𝑥,𝑡)+⋯ (34)

The next figures show the graphical representation of the approximated solution of the modified acoustic wave equation and the exact solution of the standard version of acoustic wave equation as function of space and time (contour plot and density plot of both solution). A contour plot gives essentially a topographic map of a solution and the density plot shows the values of the function at a regular erray of points and lighter region of the contour plot is higher.

From the next Figures 1, 2, 3, and 4, respectively, one can see that there are more details with the solution of the modified acoustic wave equation than in the standard solution, meaning that the details left out by neglecting the small effect of the correspondent relative density fluctuations are very important when one needs to observe the propagation of the acoustic wave through the material medium.The approximate solution of (25) has been depicted in Figures 1 and 3 and the exact solution in Figures 2 and 4.

Figure 1 

Topographic map of the solution of the modified acoustic equation for 𝑐=4, 𝑓=0.001.

Figure 2 

Topographic map of solution of acoustic wave equation for 𝑐=4 and 𝑓=0.001.

Figure 3 

Density plot of the solution of the modified acoustic wave equation.

Figure 4 

Density plot of the solution of the acoustic wave equation.

5.2. Green Function Method

To solve (25) together with (26) by means of Green function technique, one needs first to construct a suitable green function.

If G is the green function to be constructed, then G must satisfy the following equation: 𝜕²𝐺𝑥,𝑡∣𝑥₀,𝑡₀𝜕𝑥²−1𝑐²𝜕²𝐺𝑥,𝑡∣𝑥₀,𝑡₀𝜕𝑡²=−4𝜋𝛿𝑥−𝑥₀𝛿𝑡−𝑡₀. (35)

We are lucky enough, because the Green function to be constructed here is the green function of the wave equation and is given later in the case of the closed forms for the Green function for the infinite one-dimensional domain [23]: 𝐺𝑥,𝑡∣𝑥₀,𝑡₀=2𝜋𝑐𝑢𝑡−𝑡₀−𝑥−𝑥₀, (36) where 𝑢(𝑥)=0if𝑥<0,𝑢(𝑥)=1if𝑥>0. (37)

Following the Green function technique, the general solution of the modified acoustic wave equation is given later as 𝑃₁(𝑥,𝑡)=^𝑡₀^𝑥₀𝐺𝑥₁,𝑡₁∣𝑥₀,𝑡₀𝐻𝑥₁,𝑡₁𝑑𝑡₁𝑑𝑥₁, (38) where 𝐻(𝑥₁,𝑡₁) remains the same as defined earlier in Section 5.1.

6. Conclusion

In this paper, an acoustic wave equation was extended to the concept of the modified Riemann-Liouville fractional order derivative. We presented in detail some properties of the generalized Riemann-Liouville fractional order derivative approximation. We presented the analysis of the generalized equation. We highlighted the three differences between the generalized equation and the standard one. First the velocity of the sound in this case depends on time and coordinates due to the effect of the corresponding relative density fluctuations. Second the force appears due to the coordinate and time dependence of the corresponding relative density fluctuations. Third there is a derivative-free term that depends on both time and space which is proportional to 𝑃 and characterizes, depending on the coefficient sign, the retardation or enhancement propagation of acoustic waves through a material medium. The modified equation is approximately solved by using the variational iteration method and the Green function technique. The solution of the modified equation gives better prediction than the standard one.